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 θ be 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 Trigonometric Functions

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 4 Given.

Example 5 Angle θ is in standard position with its terminal side in the third quadrant. Find the exact value of cos θ if

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 θ and the horizontal axis.

Reference angles

Example 7 Finding reference angles.

Trigonometric Values of Common 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.