4.4 Trigonmetric functions of Any Angle

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# 4.4 Trigonmetric functions of Any Angle - PowerPoint PPT Presentation

4.4 Trigonmetric functions of Any Angle. Objective. Evaluate trigonometric functions of any angle Use reference angles to evaluate trig functions. Definitions of Trigonometric Functions of any Angle.

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### 4.4 Trigonmetric functions of Any Angle

Objective
• Evaluate trigonometric functions of any angle
• Use reference angles to evaluate trig functions
Definitions of Trigonometric Functions of any Angle
• Let θbe an angle in standard position with (x, y) a point on the terminal side of θ and
The cosecant function is the reciprocal of the sine.
• The secant function is the reciprocal of the cosine.
• The cotangent function is the reciprocal of the tangent function.
Example 1
• Let (-3, 4) be a point on the terminal side of θ. Find the sine, cosine, and tangent of θ.
Example 2
• Let (2, 5) be a point on the terminal side of θ. Find the sine, cosine, and tangent of θ.
Signs of the Trig Functions

A means that all trig. functions are positive.S means that all sine and cosecant functions are positive.T means that all tangent and cotangent functions are positive.C means that all cosine and secant functions are positive.

Example 3
• State whether each value is positive, negative, or zero.
• a) cos 75° positive
• b) sin 3π 0
• c) cos 5π negative
• d) sin(-3π) 0
Example 5
• Angle θis in standard position with its terminal side in the third quadrant. Find the exact value of cos θif
Example 6
• Angle θ is in standard position with its terminal side in the fourth quadrant. Find the exact value of sin θ if
Reference Angles
• Definition
• Let θ be an angle in standard position. Its reference angle is the acute angle θ’ formed by the terminal side of θand the horizontal axis.
Example 7
• Finding reference angles.
Example 8
• Use the reference angle to find sin θ, cos θ, and tan θ for each value of
Example 9
• Determine the values of θ for which
If the value of one of the trig functions of any angle is known, a calculator can be used to determine the angles having that value.
Example 10
• Find values of θ, where
• to the nearest tenth of a degree.
Example 11
• Find values of θ, where
• To the nearest hundredth of a radian.