13.2 Define General Angles and Use Radian Measure. standard position:
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1 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: Find one positive angle and one negative angle that are coterminal with the given angles..) 45 o 2.) -380 o Angles can also be measured in. There are radians in a full circle. radians = 360 o, so radians = 80 o. -To convert degrees to radians, multiply by π. 80 -To convert radians to degrees, multiply by 80. π Apr 7 0:8 AM
2 Examples:.) Convert 25 o to radians. 2.) Convert -π to degrees. 2 Degree measure Radian measure 0 o 30 o π/4 60 o π/2 2π/3 35 o 50 o 80 o 7π/6 5π/4 240 o 270 o 5π/3 35 o π/6 360 o Apr 7 0:30 AM 3.3 Evaluate Trig Functions of Any Angle Fill in the ratios using O = opposite, A = adjacent and H = hypotenuse. General Definitions of Trig Functions Let θ be an angle in standard position, and let (x,y) be the point where the terminal side of θ intersects the circle x 2 + y 2 = r 2. The six trig functions of θ are as follows: (x,y) r θ Apr 7 0:44 AM 2
3 Example: Let (-4,3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trig functions of θ. The Unit Circle The circle x 2 + y 2 =, which has center (0,0) and radius, is called the unit circle. The values of sinθ and cosθ are simply the y-coordinate and x-coordinate, respectively, of the point where the terminal side of θ intersects the unit circle. sinθ = cosθ = (x,y) θ Apr 7 0:53 AM Example Use the unit circle to evaluate the six trig functions of θ=270 o. Reference Angles Acute angles formed by the terminal side of θ and the x-axis. Recall: 30 o = 45 o = 60 o = 60 o 45 o o 45 o 3 Apr 7 :02 AM 3
4 Examples: Evaluate the six trig functions of θ. Simplify and rationalize..) θ = π/3 2.) θ = 7π/6 Apr 7 :2 AM 3.) θ=7π/4 4.) θ=2π/3 Apr 7 :5 AM 4
5 3.4 Inverse Trig Functions So far, we've learned how to evaluate trig functions of a given angle. Now, we'll study how to reverse the problem - find an angle that corresponds to a given value of a trig function. Example sinθ = Note: There are many θ's that could satisfy the above equation. For this reason, we must make some restrictions. Inverse Trig Functions: -Sine Inverse: -90 o θ 90 o -Tangent Inverse: -90 o θ 90 o Cosine Inverse: 0 o θ 80 o Apr 7 :24 AM Examples Evaluate the expression in both radians and degrees..) cos ) sin Apr 7 :40 AM 5
6 Examples Find the measure of angle θ..) 4 θ 9 2.) 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? Apr 7 :43 AM Some More Application Problems.) The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles has an angle of elevation of 30 o. The length of the escalator is 52 feet. What is the height of the escalator? 2.) A fire truck has a 00 ft. ladder whose base is 0 feet above the ground. A firefighter extends a ladder toward a burning building to reach a window 90 ft. above the ground. Draw a diagram. At what angle should the firefighter set the ladder? Apr 7 :55 AM 6
7 Homework Name: Draw an angle with the given measure in standard position..) 0 o 2.) 450 o 3.) -3π/2 (Hint: change to degrees first) Find one positive angle and one negative angle that are coterminal with the given angles. 4.) -87 o 5.) 20 o Apr 7 2:44 PM 6.) Let (-3,-4) be a point on the terminal side of an angle θ in standard position. Evaluate the six trig functions of θ. 7.) Let (-6,9) be a point on the terminal side of an angle θ in standard position. Evaluate the six trig functions of θ. Make sure to rationalize all values. Apr 7 2:50 PM 7
8 Evaluate the six trig functions of θ. Simplify and rationalize. 8.) θ = π 9.) θ = 4π/3 Apr 7 2:53 PM Evaluate the expressions in both radians and degrees. 0.) cos - (/2).) tan - (-) 2.) A crane has a 200 ft. arm with a lower end that is 5 ft. off the ground. The arm has to reach to the top of the building that is 60 ft. high. At what angle θ should the arm be set? Apr 7 2:56 PM 8
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