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Copyright © 2009 Pearson Addison-Wesley 5.3-1
5Trigonometric Functions
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5.1 Angles
5.2 Trigonometric Functions
5.3 Evaluating Trigonometric Functions
5.4 Solving Right Triangles
5 Trigonometric Functions
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Evaluating Trigonometric Functions
5.3
Right-Triangle-Based Definitions of the Trigonometric Functions ▪ Cofunctions ▪ Trigonometric Function Values of Special Angles ▪ Reference Angles ▪ Special Angles as Reference Angles ▪ Finding Function Values Using a Calculator ▪ Finding Angle Measures with Special Angles
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Right-Triangle-Based Definitions of Trigonometric Functions
For any acute angle A in standard position,
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Right-Triangle-Based Definitions of Trigonometric Functions
For any acute angle A in standard position,
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Right-Triangle-Based Definitions of Trigonometric Functions
For any acute angle A in standard position,
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Find the sine, cosine, and tangent values for angles A and B.
Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES OF AN ACUTE ANGLE
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Find the sine, cosine, and tangent values for angles A and B.
Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES OF AN ACUTE ANGLE (cont.)
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Cofunction Identities
For any acute angle A in standard position,
sin A = cos(90 A) csc A = sec(90 A)
tan A = cot(90 A) cos A = sin(90 A)
sec A = csc(90 A) cot A = tan(90 A)
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Write each function in terms of its cofunction.
Example 2 WRITING FUNCTIONS IN TERMS OF COFUNCTIONS
(a) cos 52° = sin (90° – 52°) = sin 38°
(b) tan 71° = cot (90° – 71°) = cot 19°
(c) sec 24°= csc (90° – 24°) = csc 66°
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30°- 60°- 90° Triangles
Bisect one angle of an equilateral to create two 30°-60°-90° triangles.
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30°- 60°- 90° Triangles
Use the Pythagorean theorem to solve for x.
What would x have been if the shorter side had been 5?
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Example FINDING TRIGONOMETRIC FUNCTION VALUES FOR 60°
Find the six trigonometric function values for a 60° angle.
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Example FINDING TRIGONOMETRIC FUNCTION VALUES FOR 60° (continued)
Find the six trigonometric function values for a 60° angle.
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45°- 45° Right Triangles
Use the Pythagorean theorem to solve for r.
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45°- 45° Right Triangles
Do the trig functions depend on the length of the side?
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45°- 45° Right Triangles
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Function Values of Special Angles
60
45
30
csc sec cot tan cos sin
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Reference Angles
A reference angle for an angle θ is the positive acute angle made by the terminal side of angle θ and the x-axis.
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Caution
A common error is to find the reference angle by using the terminal side of θ and the y-axis.
The reference angle is always found with reference to the x-axis.
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Find the reference angle for an angle of 218°.
Example 3(a) FINDING REFERENCE ANGLES
The positive acute angle made by the terminal side of the angle and the x-axis is 218° – 180° = 38°.
For θ = 218°, the reference angle θ′ = 38°.
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Find the reference angle for an angle of 1387°.
Example 3(b) FINDING REFERENCE ANGLES
First find a coterminal angle between 0° and 360°.
Divide 1387 by 360 to get a quotient of about 3.9. Begin by subtracting 360° three times. 1387° – 3(360°) = 307°.
The reference angle for 307° (and thus for 1387°) is 360° – 307° = 53°.
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Find the values of the six trigonometric functions for 210°.
Example 4 FINDING TRIGONOMETRIC FUNCTION VALUES OF A QUADRANT III ANGLE
The reference angle for a 210° angle is 210° – 180° = 30°.
Choose point P on the terminal side of the angle so the distance from the origin to P is 2.
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Example 4 FINDING TRIGONOMETRIC FUNCTION VALUES OF A QUADRANT III ANGLE (continued)
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Find the exact value of cos (–240°).
Example 5(a) FINDING TRIGONOMETRIC FUNCTION VALUES USING REFERENCE ANGLES
Since an angle of –240° is coterminal with an angle of –240° + 360° = 120°, the reference angle is 180° – 120° = 60°.
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Find the exact value of tan 675°.
Example 5(b) FINDING TRIGONOMETRIC FUNCTION VALUES USING REFERENCE ANGLES
Subtract 360° to find a coterminal angle between 0° and 360°: 675° – 360° = 315°.
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Caution
When evaluating trigonometric functions of angles given in degrees, remember that the calculator must be set in degree mode.
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Example 6 FINDING FUNCTION VALUES WITH A CALCULATOR
(a) sin 53°
Approximate the value of each expression.
(b) sec 97.977°
sec 97.977° ≈ –7.20587921
Calculators do not have a secant key, so first find cos 97.977° and then take the reciprocal.
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Example 6 FINDING FUNCTION VALUES WITH A CALCULATOR (continued)
(c)
Approximate the value of each expression.
Use the reciprocal identity
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Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition.
Example 7 USING INVERSE TRIGONOMETRIC FUNCTIONS TO FIND ANGLES
a) cos θ = .5 We know that cos ____ = .5 To find this on the calculator we use θ =
b) Find θ if cos θ = .9211854056
c) Find θ if sec θ = 1.2228
____5.0cos 1
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Find all values of θ, if θ is in the interval [0°, 360°)
and
Example 8 FINDING ANGLE MEASURES GIVEN AN INTERVAL AND A FUNCTION VALUE
Since cos θ is negative, θ must lie in quadrant II or III.
The angle in quadrant II is 180° – 45° = 135°.
The angle in quadrant III is 180° + 45° = 225°.
The absolute value of cos θ is so the reference angle is 45°.
Example 9
Problem 141, page 529
When highway curves are designed, the outside of the curve is often slightly elevated or inclined above the inside of the curve. This inclination is called superelevation . For safety reasons, it is important that both the curve’s radius and superelevation are correct for a given speed limit. If an automobile is traveling at velocity V (in feet per second), the safe radius R for a curve with superelevation θ is modeled by the formula where f and g are constants
)tan(
2
fg
VR
Example 9
)tan(
2
fg
VR
a) A roadway is being designed for automobiles traveling at 45mph.If θ = 3°, g = 32.2, and f = 0.14, calculate R to the nearest foot. (Hint: You will need to know that 1 mile = 5280 feet)
b) A highway curve has radius R = 1150 ft and a superelevation ofθ = 2.1°. What should the speed limit (in miles per hour) be for this curve? Use the same values for f and g from part (a).