Section 4.4 Trigonometric Functions of Any Angle

Definition 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 r is the distance from the origin to (x, y). r (x, y) Θ y x

Using the information from the previous slide and your knowledge about trig functions, write the six trig functions’ ratios in terms of x, y, and r.

r (x, y) Θ y x

Example 1

Let (-2, 3) be a point on the terminal side of Θ. Find the six trig functions.

On the next slide determine the sign of the trigonometric functions in each quadrant.

Quadrant I sin θ cos θ tan θ Quadrant III sin θ cos θ tan θ Quadrant II sin θ cos θ tan θ Quadrant IV sin θ cos θ tan θ − − + − − − − +

Example 2 What quadrant is θ in? The quadrant that sine is positive and tangent is negative is the 2 nd quadrant.

Example 3 Let θ be an angle in Quadrant III such that Find sec θ and tan θ.

Drawing a right triangle with the terminal side of θ as the hypotenuse find the legs of the right triangle

Example 4 The terminal side of θ lies on the given line 2x – y = 0 in the 3 rd quadrant. Find the values of sine, cosine, and tangent of θ by finding a point on the line in the specified quadrant.

All points in the 3 rd quadrant have both coordinates negative. To find a point on this line pick a negative x-coordinate. I am picking x = -2. Solve for y. 2(-2) – y = 0 -4 – y = 0 y = -4 My point is (-2, -4).

HW: pp (2-36 even)

Quadrant angles are angle measures on the axes. θ = 0 °, 90 °, 180 °, 270 ° When finding the six trig functions of the quadrant angles, let r = 1 and then find (x, y). (1, 0) (0, 1) (-1, 0) (0, -1) 0° 90° 180° 270°

Example 5 Evaluate the six trig functions when a.θ = π

The values of trigonometric functions of angles greater than 90 ° or less than 0° can be determined from their values at corresponding ACUTE ANGLES called reference angles. This is the angle θ’ formed by the terminal side of θ and the nearest horizontal axis. You find the reference angle always from the HORIZONTAL AXIS. All angles in the 1 st quadrant are reference angles of themselves.

Quadrant II Θ Θ’Θ’ In radians Θ’ = In degrees Θ’ = π – Θ180° – Θ

Quadrant III Θ Θ’Θ’ In radians Θ’ =In degrees Θ’ =Θ – π Θ – 180°

Quadrant IV Θ Θ’Θ’ In radians Θ’ = In degrees Θ’ = 2π – Θ 360° – Θ

Example 6 Find the reference angle θ’: a.300° c.213° e.1.7 rad.

a.300° 300° is in the 4 th quadrant. θ’ = 360 – 300 = 60° Hint: Change into a positive angle. This angle is in the 2 nd quadrant.

c.213° θ’ = 213 – 180 = 33° e.1.7 rad. θ’ = π – 1.7 = 3.14 – 1.7 = 1.44 rad. 3 rd quadrant 2 nd quadrant

For the following add a positive or negative sign to the right side of the equation. For θ in the 2 nd Quadrant: sin θ = __sin θ’ cos θ = __cos θ’ For θ in the 3 rd Quadrant: sin θ = __sin θ’ cos θ = __cos θ’ Do the same thing for θ in the 4 th Quadrant. − + − −

Trig Functions of Special Angles Special angles in trigonometry are angles whose reference angles are 30°, 45°, or 60°, and all quadrant angles in degrees. In radians the special angles are angles whose reference angles are and all quadrant angles. The trig functions values for these angles will always be exact!!!!!

Example 7 Find sine, cosine, and tangent of the angle without a calculator. a.750°

Find the coterminal angle that is 0° < θ < 360°. 750 – 360 = 390 – 360 = 30°

Find the coterminal angle that is 0 < θ < 2π.

Example 8 Find two solutions for the equation. Give your answers in degrees (0° ≤ θ < 360°) and in radians (0 ≤ θ < 2π).

The sine function is positive in the 1 st and 2 nd quadrants.

The tangent function is negative in the 2 nd and 4 th quadrants. HW: pp (38-86 even, 90, 92)