Trigonometry LESSON ONE - Degrees and Radians Lesson Notes

Transcription

1 8 = 6 Trigonometry LESSON ONE - Degrees and Radians Example : Define each term or phrase and draw a sample angle. Angle in standard position. b) Positive and negative angles. Draw. c) Reference angle. d)draw the first positive e) Principal angle. f)find the first four positive Conversion Multiplier Reference Chart (Example ) degree radian revolution degree radian revolution Example : Three Angle Types: Degrees, Radians, and Revolutions. i. Define degrees. ii. Define radians. iii. Define revolutions. b) Use conversion multipliers to answer the questions and fill in the reference chart. i.ii.iii.iv.v.vi. c) Contrast the decimal approximation of a radian with the exact value of a radian. i.(decimal approximation). ii.(exact value). Example : Convert each angle to the requested form. Round all decimals to the nearest hundredth. b) c)d) to degrees. e)to degrees. f)as an approximate radian decimal. g) h)to degrees. i)to radians. Example 4: The diagram shows commonly used degrees. When complete, memorize the diagram. Method One: a conversion multiplier. b) Method Two: Use a shortcut (counting radians). Example 5: Draw each of the following angles in standard position. State the reference angle. b)c)d)e) Example 6: Draw each of the following angles in standard position. State the principal and reference angles. b)c) 9 d)

2 LESSON ONE - Degrees and Radians Example 7: b) c), d) Example 8: For each angle, use estimation to find the principal angle. b)c)d) Example 9: p c b) p (Find c c) c n and p ) d) c (Find n and p ) Example : In addition to the three primary trigonometric ratios (sin, cos, and tan), there are three reciprocal ratios (csc, sec, and cot). Given a triangle with side lengths of x and y, and a hypotenuse of length r, the six trigonometric ratios are as follows: r x y sin y r cos x r tan y x csc sec cot sin cos tan r y r x x y in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle. b) in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle. Example : Determine the sign of each trigonometric ratio in each quadrant. sinb)c)d)e)f) g) How do the quadrant signs of the reciprocal trigonometric ratios (csc, sec, and cot) compare to the quadrant signs of the primary trigonometric ratios (sin, cos, and tan)? Example : Given the following conditions, find the quadrant(s) where the angle could potentially exist. i. sinii. cos iii. tan > b) i. sincos ii. sectan iii. csccot c) i. sincscii. cos and csc iii. sectan Example : Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a radian. Example 4: Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a degree. b)

3 Trigonometry LESSON ONE - Degrees and Radians Example 5: Calculating with a calculator. When you solve a trigonometric equation in your calculator, the answer you get for can seem unexpected. Complete the following chart to learn how the calculator processes your attempt to solve for. b) find the reference angle using a sine ratio, Jordan tries to find it using a cosine ratio, and Dylan tries to find it using a tangent ratio. Why does each person get a different result from their calculator? Mark s Calculation of sin = 5 If the angle could exist in either quadrant or... I or II I or III I or IV II or III II or IV III or IV Jordan s Calculation of cos = -4 5 The calculator always picks quadrant Dylan s Calculation of tan = -4 Example 6: The formula for arc length is a, where a is the arc length, is the central angle in radians, and r is the radius of the circle. The radius and arc length must have the same units. b) e) r n a Example 7: Area of a circle sector. Derive the formula for the area of a circle sector, r. Find the area of each shaded region. b) e) 9 cm

4 LESSON ONE - Degrees and Radians Example 8: The formula for angular speed is, where Calculate the requested quantity in each scenario. Round all decimals to the nearest hundredth. Calculate the angular speed in degrees per second. b) c) in one second? d) e) angular speed of one of the bicycle wheels and express the answer using revolutions per second. Example 9: Calculate the angular speed of the satellite. b)

5 (cos, sin) Trigonometry LESSON TWO - The Unit Circle Example : Introduction to Circle Equations. A circle centered at the origin can be represented by the relation x + y = r, where r is the radius of the circle. Draw each circle: i. x + y = 4 ii. x + y = b) A circle centered at the origin with a radius of has the equation x + y =. This special circle is called the unit circle. Draw the unit circle and determine if each point exists on the circumference of the unit circle: i. (.6,.8) ii. (.5,.5) - - c) Using the equation of the unit circle, x + y =, find the unknown coordinate of each point. Is there more than one unique answer? - - i. ii., quadrant II. iii. (-, y) iv., cos >. Example : The following diagram is called the unit circle. Commonly used angles are shown as radians, and their exact-value coordinates are in brackets. Take a few moments to memorize this diagram. When you are done, use the blank unit circle on the next page to practice drawing the unit circle from memory. What are some useful tips to memorize the unit circle? b) Draw the unit circle from memory. Example : Use the unit circle to find the exact value of each expression. sin b) cos 8 c) cos 6 4 d) sin 6 e) sin f) cos g) sin h) cos - Example 4: Use the unit circle to find the exact value of each expression. cos 4 b) -cos c) sin 6 d) cos e) sin f) -sin g) cos 4 (-84 ) h) cos

6 LESSON TWO - The Unit Circle (cos, sin) Example 5: The unit circle contains values for cos and sin only. The other four trigonometric ratios can be obtained using the identities on the right. Find the exact values of sec and csc in the first quadrant. Example 6: Find the exact values of tan and cot in the first quadrant. sec = cos tan = sin cos csc = cot = sin tan = cos sin Example 7: Use symmetry to fill in quadrants II, III, and IV for sec, csctan and cot. Example 8: Find the exact value of each expression. sec b) sec c) csc d) csc 4 e) tan 6 f) -tan g) cot 4 (7 ) h) cot 6 Example 9: Find the exact value of each expression. b) Example : Find the exact value of each expression. Example : Find the exact value of each expression. Example : Verify each trigonometric statement with a calculator. Note: Every question in this example has already been seen earlier in the lesson. c) d) e) f) g) h)

7 (cos, sin) Trigonometry LESSON TWO - The Unit Circle Example : Coordinate Relationships on the Unit Circle What is meant when you are asked to find on the unit circle? b) Find one positive and one negative angle such that c) How does a half-rotation around the unit circle change the coordinates? 6 d) How does a quarter-rotation around the unit circle change the coordinates? e) What are the coordinates of P()? Express coordinates to four decimal places. Example 4: Circumference and Arc Length of the Unit Circle What is the circumference of the unit circle? b) How is the central angle of the unit circle related to its corresponding arc length? c) If a point on the terminal arm rotates from (, ) A Diagram for Example 4 (d). to, what is the arc length? d) What is the arc length from point A to point B on the unit circle? B Example 5: Domain and Range of the Unit Circle Is sin = possible? Explain, using the unit circle as a reference. b) Which trigonometric ratios are restricted to a range of? Which trigonometric ratios exist outside that range? c) If exists on the unit circle, how can the unit circle be used to find cos? How many values for cos are possible? d) If exists on the unit circle, how can the equation of the Chart for Example 5 (b). Range Number Line cossin cscsec tan unit circle be used to find sin? How many values for sin are possible? e) If cossinare possible?

8 LESSON TWO - The Unit Circle (cos, sin) Example 6: Unit Circle Proofs Use the Pythagorean Theorem to prove that the equation of the unit circle is x + y =. b) Prove that the point where the terminal arm intersects the unit circle, P(), has coordinates of (cos, sin). c) If the point exists on the terminal arm of a unit circle, find the exact values of the six trigonometric ratios. State the reference angle and standard position angle to the nearest hundredth of a degree. Example 7: In a video game, the graphic of a butterfly needs to be rotated. To make the butterfly graphic rotate, the programmer uses the equations: to transform each pixel of the graphic from its original coordinates, (x, y), to its new coordinates, (x, y ). Pixels may have positive or negative coordinates. If a particular pixel with coordinates of (5, ) is rotated by, what are the new 6 coordinates? Round coordinates to the nearest whole pixel. b) If a particular pixel has the coordinates (64, 48) after a rotation of, what were the 4 original coordinates? Round coordinates to the nearest whole pixel. Example 8: From the observation deck of the Calgary Tower, an observer has to A B down to see point B. Show that the height of the observation x deck is h =. cot A - cot B A B b) If A =, B =, and x =.9 m, how high is the observation deck above the ground, to the nearest metre? h A B x

9 y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example : Label all tick marks in the following grids and state the coordinates of each point. y 5 y -5 y - y Example : Exploring the graph of y = sin Draw y = sinb) State the amplitude. c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. f) State the -intercepts. Write your answer using a general form expression. g) State the y-intercept. h) State the domain and range. Example : Exploring the graph of y = cos Draw y = cosb) State the amplitude. c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. f) State the -intercepts. Write your answer using a general form expression. g) State the y-intercept. h) State the domain and range. Example 4: Exploring the graph of y = tan Draw y = tanb) Is it correct to say a tangent graph has an amplitude? c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. f) State the -intercepts. Write your answer using a general form expression. g) State the y-intercept. h) State the domain and range.

10 LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example 5: The a Parameter. Graph each function over the domain. y = sin b) y = -cos c) y = sin 5 d) y = cos Example 6: The a Parameter. Determine the trigonometric function corresponding to each graph. write a sine function. b) write a sine function. c) write a cosine function. d) write a cosine function ( ) Example 7: The d Parameter. Graph each function over the domain. y = sin- b) y = cos+ 4 c) y = - sin+ d) y = cos- Example 8: The d Parameter. Determine the trigonometric function corresponding to each graph. write a sine function. b) write a cosine function. c) write a cosine function. d) write a sine function Example 9: The b Parameter. Graph each function over the stated domain. y = cos b) y = sin c) y = cos d) y = sin 5 Example : The b Parameter. Graph each function over the stated domain. y = -sin( (- b) y = 4cos+ 6 (- c) y = cos - (- 4 d) y = sin

11 y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example : The b Parameter. Determine the trigonometric function corresponding to each graph. write a cosine function. b) write a sine function. c) write a cosine function. d) write a sine function Example : The c Parameter. Graph each function over the stated domain. (-4 b) (-4 c) (- d) (- Example : The c Parameter. Graph each function over the stated domain. b) (- c) (- d) (- Example 4: The c Parameter. Determine the trigonometric function corresponding to each graph. write a cosine function. b) write a sine function. c) write a sine function. d) write a cosine function Example 5: a, b, c, & d Parameters. Graph each function over the stated domain. -4 b) c) - d)

12 LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example 6: a, b, c, & d. Determine the trigonometric function corresponding to each graph. write a cosine function. b) write a cosine function. - Example 7: Exploring the graph of y = sec - y Draw y = secb) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. e) Given the graph of f() = cos - - Example 8: Exploring the graph of y = csc y Draw y = cscb) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. e) Given the graph of f() = sin Example 9: Exploring the graph of y = cot - - y Draw y = cotb) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. e) Given the graph of f() = tan - - Example : Graph each function over the domain. State the new domain and range. - y = sec - y = sec - y = csc - y = cot

13 h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example : Trigonometric Functions of Angles i. Graph: ( < b) i. Graph: (º < 54º) ii. Graph this function using technology. ii. Graph this function using technology. Example : Trigonometric Functions of Real Numbers. i. Graph: b) i. Graph: ii. Graph this function using technology. ii. Graph this function using technology. c) What are three differences between trigonometric functions of angles and trigonometric functions of real numbers? Example : Determine the view window for each function and sketch each graph. b) Example 4: Determine the view window for each function and sketch each graph. b) Example 5: Determine the trigonometric function corresponding to each graph. write a cosine function. b) write a sine function. c) write a cosine function. d) write a sine function. 5 (8, 9) (45, 5) (6, -) (, -5) Example 6: If the transformation g( b) Find the range of 4. c) If the range of y = cos + d is [-4, k], determine the values of d and k. d) State the range of f() - = msin() + n. e) The graphs of f() and g() intersect at the points and. If the amplitude of each graph is quadrupled, determine the new points of intersection.

14 LESSON FOUR - Trigonometric Functions II h(t) t Example 7: If the point lies on the graph of, find the value of a. b) Find the y-intercept of. c) The graphs of f() and g() intersect at the point (m, n). Find the value of f(m) + g(m). d) The graph of f( about the x-axis. n k Graph for Example 7c (m, n) m Graph for Example 7d g() f() f() If the point exists on the graph b g() of g(state the vertical stretch factor. Example 8: The graph shows the height of a pendulum bob as a function of time. One cycle of a pendulum consists of two swings - a right swing and a left swing. Write a function that describes the height of the pendulum bob as a function of time. b) If the period of the pendulum is halved, how will this change the parameters in the function you wrote in part (? c) If the pendulum is lowered so its lowest point is cm above the ground, how will this change the parameters in the function you wrote in part (? ground level h(t) cm 8 cm 4 cm cm Graph for Example 8 s s s 4 s t Example 9: A wind turbine has blades that are m long. An observer notes that one blade makes complete rotations (clockwise) every minute. The highest point of the blade during the rotation is 5 m. Using Point A as the starting point of the graph, draw the height of the blade over two rotations. b) Write a function that corresponds to the graph. c) Do we get a different graph if the wind turbine rotates counterclockwise? A Example : A person is watching a helicopter ascend from a distance 5 m away from the takeoff point. Write a function, h(), that expresses the height as a function of the angle of elevation. Assume the height of the person is negligible. b) Draw the graph, using an appropriate domain. c) Explain how the shape of the graph relates to the motion of the helicopter. 5 m h

15 h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example : A mass is attached to a spring 4 m above the ground and allowed to oscillate from its equilibrium position. The lowest position of the mass is.8 m above the ground, and it takes s for one complete oscillation. Draw the graph for two full oscillations of the mass. b) Write a sine function that gives the height of the mass above the ground as a function of time. c) Calculate the height of the mass after. seconds. Round your answer to the nearest hundredth. d) In one oscillation, how many seconds is the mass lower than. m? Round your answer to the nearest hundredth. Example : A Ferris wheel with a radius of 5 m rotates once every seconds. Riders board the Ferris wheel using a platform m above the ground. Draw the graph for two full rotations of the Ferris wheel. b) Write a cosine function that gives the height of the rider as a function of time. c) Calculate the height of the rider after.6 rotations of the Ferris wheel. Round your answer to the nearest hundredth. d) In one rotation, how many seconds is the rider higher than 6 m? Round your answer to the nearest hundredth. Example : The following table shows the number of daylight hours in Grande Prairie. December March June September December 6h, 46m h, 7m 7h, 49m h, 7m 6h, 46m Convert each date and time to a number that can be used for graphing. Day Number December = March = June = September = December = Daylight Hours 6h, 46m = h, 7m = 7h, 49m = h, 7m = h, 46m = b) Draw the graph for one complete cycle (winter solstice to winter solstice). c) Write a cosine function that relates the number of daylight hours, d, to the day number, n. d) How many daylight hours are there on May? Round your answer to the nearest hundredth. e) In one year, approximately how many days have more than 7 daylight hours? Round your answer to the nearest day.

16 LESSON FOUR - Trigonometric Functions II h(t) t Example 4: The highest tides in the world occur between New Brunswick and Nova Scotia, in the Bay of Fundy. Each day, there are two low tides and two high tides. The chart below contains tidal height data that was collected over a 4-hour period. Time Decimal Hour Height of Water (m) Convert each time to a decimal hour. b) Graph the height of the tide for one full cycle (low tide to low tide). c) Write a cosine function that relates Low Tide High Tide Low Tide High Tide : AM 8: AM : PM 8: PM the height of the water to the elapsed time. d) What is the height of the water at 6:9 AM? Round your answer to the nearest hundredth. e) For what percentage of the day is the height of the water greater than m? Round your answer to the nearest tenth. Example 5: A wooded region has an ecosystem that supports both owls and mice. Owl and mice populations vary over time according to the equations: Bay of Fundy Bay of Fundy Note: Actual tides at the Bay of Fundy are 6 hours and minutes apart due to daily changes in the position of the moon. In this example, we will use 6 hours for simplicity. Owl population: Mouse population: where O is the population of owls, M is the population of mice, and t is the time in years. Graph the population of owls and mice over six years. b) Describe how the graph shows the relationship between owl and mouse populations. Example 6: The angle of elevation between the 6: position and the : position of a historical building s clock, as measured from an observer standing on a hill, is The observer also knows that he is standing 44 m away from the clock, and his eyes are at the same height as the base of the clock. The radius of the clock is the same as the length of the minute hand. If the height of the minute hand s tip is measured relative to the bottom of the clock, what is the height of the tip at 5:8, to the nearest tenth of a metre? Example 7: Shane is on a Ferris wheel, and his height can be described by the equation. Tim, a baseball player, can throw a baseball with a speed of m/s. If Tim throws a ball directly upwards, the height can be determined by the equation h ball (t) = -4.95t + t +. If Tim throws the baseball 5 seconds after the ride begins, when are Shane and the ball at the same height? m

17 LESSON FIVE - Trigonometric Equations Example : Primary Ratios. Find all angles in the domain that satisfy the given equation. Write the general solution. Solve equations non-graphically using the unit circle. b) c) d) tan Example : Primary Ratios. Find all angles in the domain that satisfy the given equation. Write the general solution. Solve equations graphically with intersection points. b) c) d) e) f) undefined Example : Primary Ratios. Find all angles in the domain 6 that satisfy the given equation. Write the general solution. Solve equations non-graphically with a calculator (degree mode). c) Example 4: Primary Ratios. Find all angles in the domain that satisfy the given equation. Solve equations graphically with -intercepts. b) Example 5: Primary Ratios. Solve - cos feature of a calculator. Example 6: Primary Ratios. cos feature of a calculator. b) using the unit circle. b) using the unit circle. Example 7: Reciprocal Ratios. Find all angles in the domain that satisfy the given equation. Write the general solution. Solve equations non-graphically using the unit circle. c) c) c) d) using d) using Example 8: Reciprocal Ratios. Find all angles in the domain that satisfy the given equation. Write the general solution. Solve equations graphically with intersection points. b) c) d) e) f) Example 9: Reciprocal Ratios. Find all angles in the domain 6 that satisfy the given equation. Write the general solution. Solve non-graphically with a calculator (degree mode). c) Example : Reciprocal Ratios. Find all angles in the domain that satisfy the given equation. Write the general solution. Solve equations graphically with -intercepts. θ b) θ

18 LESSON FIVE - Trigonometric Equations Example : Reciprocal Ratios. cos feature of a calculator. Example : Reciprocal Ratios. cos feature of a calculator. Solve csc b) using the unit circle. Solve sec c) 6 d) using Example : First-Degree Trigonometric Equations. Find all angles in the domain that satisfy the given equation. Write the general solution. b) using the unit circle. c) c) d) Example 4: First-Degree Trigonometric Equations. Find all angles in the domain that satisfy the given equation. Write the general solution. Example 5: Second-Degree Trigonometric Equations. Find all angles in the domain that satisfy the given equation. Write the general solution. sin b) 4cos c) cos d) tan 4 Example 6: Second-Degree Trigonometric Equations. Find all angles in the domain that satisfy the given equation. Write the general solution. sin b) csc c) sin Example 7: Double and Triple Angles. b) Example 8: Half and Quarter Angles. 4 b) 8 d) using Example 9: b) Example : d. b) c) of rotation (in degrees)? d

19 LESSON SIX - Trigonometric Identities I Example : Understanding Trigonometric Identities. Why are trigonometric identities considered to be a special type of trigonometric equation? b) Which of the following trigonometric equations are also trigonometric identities? i. ii. iii. iv. v. Example : The Pythagorean Identities. Using the definition of the unit circle, derive the identity sin x + cos x =. Why is sin x + cos x = called a Pythagorean Identity? b) Verify that sin x + cos x = is an identity using (i) x = and (ii) x =. c) Verify that sin x + cos x = is an identity using a graphing calculator to draw the graph. d) Using the identity sin x + cos x =, derive + cot x = csc x and tan x + = sec x. e) Verify that + cot x = csc x and tan x + = sec x are identities for x =. f) Verify that + cot x = csc x and tan x + = sec x are identities graphically. Example : Reciprocal Identities. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Example 4: Reciprocal Identities. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Example 5: Pythagorean Identities. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Example 6: Pythagorean Identities. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

20 LESSON SIX- Trigonometric Identities I Example 7: Common Denominator Proofs. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. c) d) Example 8: Common Denominator Proofs. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Example 9: Assorted Proofs. Prove each identity. For simplicity, ignore NPV s and graphs. Example : Assorted Proofs. Prove each identity. For simplicity, ignore NPV s and graphs. b) Example : Assorted Proofs. Prove each identity. For simplicity, ignore NPV s and graphs. Example : Exploring the proof of Prove algebraically that. b) Verify that for. c) State the non-permissible values for. d) Show graphically that. Are the graphs exactly the same?

21 LESSON SIX - Trigonometric Identities I Example : Exploring the proof of. Prove algebraically that. b) Verify that for. c) State the non-permissible values for. d) Show graphically that. Are the graphs exactly the same? Example 4: Exploring the proof of Prove algebraically that b) Verify that for. c) State the the non-permissible values for. d) Show graphically that. Are the graphs exactly the same? Example 5: Equations with Identites. Example 6: Equations with Identites. Example 7: Equations with Identites. b) c) d)

22 LESSON SIX- Trigonometric Identities I Example 8: Use the Pythagorean identities to find the indicated value and draw the corresponding triangle. If the value of find the value of cosx within the same domain. b) If the value of, find the value of seca within the same domain. 7 c) cos =, and cot <, 7 Example 9: Trigonometric Substitution. Using the triangle to the right, show that can be expressed as. Hint: Use the triangle to find a trigonometric expression equivalent to b. a b b) Using the triangle to the right, show that can be expressed as. Hint: Use the triangle to find a trigonometric expression equivalent to a. a 4

23 LESSON SEVEN - Trigonometric Identities II Example : Evaluate each trigonometric sum or difference. c) d) e) f) Example : Write each expression as a single trigonometric ratio. c) Example : Find the exact value of each expression. Given the exact values of cosine and sine for 5, fill in the blanks for the other angles. Example 4: Find the exact value of each expression. Example d b) c) Example 5: Double-angle identities. Prove the double-angle sine identity, sinx = sinxcosx. b) Prove the double-angle cosine identity, cosx = cos x - sin x. c) The double-angle cosine identity, cosx = cos x - sin x, can be expressed as cosx = - sin x or cosx = cos x -. Derive each identity. d) Derive the double-angle tan identity,. Example 6: Double-angle identities. Evaluate each of the following expressions using a double-angle identity. i. ii. b) Express each of the following expressions using a double-angle identity. iii. i. ii. iii. iv. c) Write each of the following expression as a single trigonometric ratio using a double-angle identity. i. ii. iii. iv.

24 LESSON SEVEN- Trigonometric Identities II Example 7: Prove each trigonometric identity. b) Note: Variable restrictions may be ignored for the proofs in this lesson. Example 8: Prove each trigonometric identity. c) d) Example 9: Prove each trigonometric identity. Example : Prove each trigonometric identity. b) Example : Prove each trigonometric identity. c) d) Example : Prove each trigonometric identity.

25 LESSON SEVEN - Trigonometric Identities II Example : Prove each trigonometric identity. b) c) d) Example 4: b) Example 5: Example 6: Diagram for Example 8 A c) b) d) B Example 7: C Example 8: Trigonometric identities and geometry. Show that b) If A = and B = 89, what is the value of C? x 57 Diagram for Example 9 Example 9: Trigonometric identities and geometry. Solve for x. Round your answer to the nearest tenth B A 5