13.4 Chapter 13: Trigonometric Ratios and Functions. Section 13.4

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1 13.4 Chapter 13: Trigonometric Ratios and Functions Section

2 13.4 Chapter 13: Trigonometric Ratios and Functions Section

3 Key Concept Section

4 Key Concept Section

5 Key Concept Section

6 EXAMPLE 1 Evaluate inverse trigonometric functions Evaluate the expression in both radians and degrees. a. cos b. sin 1 2 c. tan 1 ( 3 ) Section

7 EXAMPLE 1 Evaluate inverse trigonometric functions Evaluate the expression in both radians and degrees. a. cos SOLUTION a. When 0 θ π or 0 θ 180, the angle whose cosine is 3 2 θ = cos π = cos 1 3 θ = = Section

8 EXAMPLE 1 Evaluate inverse trigonometric functions Evaluate the expression in both radians and degrees. b. sin 1 2 SOLUTION b. There is no angle whose sine is 2. So, sin 1 2 is undefined. Section

9 EXAMPLE 1 Evaluate inverse trigonometric functions Evaluate the expression in both radians and degrees. c. tan 1 ( 3 ) SOLUTION c. When π < θ < π, or 90 < θ < 90, the 2 2 angle whose tangent is 3 is: θ = tan 1 ( 3 ) π = θ = tan ( ) = 60 Section

10 EXAMPLE 2 Solve a trigonometric equation 5 Solve the equation sinθ= 8 where 180 < θ < 270. Section

11 EXAMPLE 2 Solve a trigonometric equation 5 Solve the equation sinθ= 8 where 180 < θ < 270. SOLUTION STEP 1 Use a calculator to determine that in the interval 90 θ 90, the angle whose sine is 5 is sin This 8 8 angle is in Quadrant IV, as shown. Section

12 EXAMPLE 2 Solve a trigonometric equation STEP 2 Find the angle in Quadrant III (where 180 < θ < 270 ) that has the same sine value as the angle in Step 1. The angle is: θ = CHECK : Use a calculator to check the answer. 5 sin = 8 Section

13 GUIDED PRACTICE for Examples 1 and 2 Evaluate the expression in both radians and degrees. 1. sin ANSWER π 4, cos ANSWER π 3, tan 1 ( 1) ANSWER π 4, 45 Section

14 GUIDED PRACTICE for Examples 1 and 2 Evaluate the expression in both radians and degrees. 4. sin 1 ( 1 ) 2 ANSWER π 6, 30 Section

15 GUIDED PRACTICE for Examples 1 and 2 Solve the equation for 5. cos θ = 0.4; 270 < θ < 360 ANSWER about tan θ = 2.1; 180 < θ < 270 ANSWER about sin θ = 0.23; 270 < θ < 360 ANSWER about Section

16 GUIDED PRACTICE for Examples 1 and 2 Solve the equation for 8. tan θ = 4.7; 180 < θ < 270 ANSWER about sin θ = 0.62; 90 < θ < 180 ANSWER about cos θ = 0.39; 180 < θ < 270 ANSWER about Section

17 EXAMPLE 3 Standardized Test Practice Section

18 EXAMPLE 3 Standardized Test Practice SOLUTION In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse, so use the inverse cosine function to solve for θ. cosθ = adj hyp = 6 11 θ = cos ANSWER The correct answer is C. Section

19 EXAMPLE 4 Write and solve a trigonometric equation Monster Trucks A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle θ of the ramp? Section

20 EXAMPLE 4 Write and solve a trigonometric equation SOLUTION STEP 1 Draw: a triangle that represents the ramp. STEP 2 Write: a trigonometric equation that involves the ratio of the ramp s height and horizontal length. tanθ = opp adj = 8 20 Section

21 EXAMPLE 4 Write and solve a trigonometric equation STEP 3 Use: a calculator to find the measure of θ. θ = tan ANSWER The angle of the ramp is about 22. Section

22 GUIDED PRACTICE for Examples 3 and 4 Find the measure of the angle θ. 11. SOLUTION In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse. So, use the inverse cosine function to solve for θ. cosθ = adj hyp = 4 9 = 63.6 θ cos Section

23 GUIDED PRACTICE for Examples 3 and 4 Find the measure of the angle θ. 12. SOLUTION In the right triangle, you are given the lengths of the side opposite to θ and the side adjacent. So, use the inverse tan function to solve for θ. tanθ = opp adj = 10 8 θ = tan Section

24 GUIDED PRACTICE for Examples 3 and 4 Find the measure of the angle θ. 13. SOLUTION In the right triangle, you are given the lengths of the side opposite to θ and the hypotenuse. So, use the inverse sin function to solve for θ. sinθ = opp hyp = 5 12 θ = sin Section

25 GUIDED PRACTICE for Examples 3 and WHAT IF? In Example 4, suppose a monster truck drives 26 feet on a ramp before jumping onto a row of cars. If the ramp is 10 feet high, what is the angle θ of the ramp? SOLUTION STEP 1 STEP 2 Draw: a triangle that represents the ramp. Write: a trigonometric equation that involves the ratio of the ramp s height and horizontal length. tanθ = opp adj = Section

26 GUIDED PRACTICE for Examples 3 and 4 STEP 3 Use: a calculator to find the measure of θ. θ = tan ANSWER The angle of the ramp is about Section

27 HOMEWORK Sec 13-4 (pg 878) 3-27 every 3 rd Section

13.2 Define General Angles and Use Radian Measure. standard position:

13.2 Define General Angles and Use Radian Measure. standard position: 3.2 Define General Angles and Use Radian Measure standard position: Examples: Draw an angle with the given measure in standard position..) 240 o 2.) 500 o 3.) -50 o Apr 7 9:55 AM coterminal angles: Examples: