Section 7-5

1. Section 7-5 The Other Trigonometric Functions

2. Copy and complete the chart:

3. The other Trig Functions We can define the other four trigonometric functions of an angle Θ in terms of the x- and y-coordinates of a point on the terminal ray of Θ.

4. The other Trig Functions Tangent of Θ: tan Θ = Cotangent of Θ: cot Θ = Secant of Θ: sec Θ = Cosecant of Θ: csc Θ =

5. The other Trig Functions Since cos Θ = and sin Θ = , we can write these four new functions in terms of cos Θ and sin Θ. Notice that sec Θ and cos Θ are reciprocals, as are csc Θ and sin Θ. This is also true of cot Θ and tan Θ:

6. The other Trig Functions The signs of these functions in the various quadrants are summarized in the table below:

7. The other Trig Functions You can remember the signs by remembering “All Students Take Calculus.” All trig functions positive Sine and Cosecant are positive Students All Cosine and Secant are positive Take Tangent and Cotangent are positive Calculus

9. The Tangent Graph Imagine on the unit circle a particle P that starts at (1, 0) and rotates counterclockwise around the origin. Every position (x, y) corresponds to an angle Θ where tan Θ = .

10. The Tangent Graph As the particle moves through the first quadrant: When P is at (1, 0), Θ = 0 and tan Θ = . As P moves toward (0, 1), y increases and x decreases, so that tan Θ = gets larger. Although the definition of tangent would suggest that tan this expression is undefined, so we say that tan is also undefined. As the particle continues to move around the unit circle, we can analyze the other values of tan Θ and obtain the tangent graph shown on p. 284. Notice that the graph has a vertical asymptotes at odd multiples of . Also notice that the tangent function is periodic with fundamental period π (or 180°).

11. Tangent Graph

12. Cotangent Graph

13. The Secant Graph Since the secant function is the reciprocal of the cosine function, we can obtain the secant graph (shown in red on p. 284) using the cosine graph (black) and these facts: sec Θ = 1 when cos Θ = 1: at Θ = 0, ±2π, ±4 π, … sec Θ = -1 when cos Θ = -1: at Θ = ± π, ±3 π, ±5 π … sec Θ is undefined when cos = 0: at Θ = |sec Θ| gets larger as |cos Θ| gets smaller

14. The Secant Graph Notice that the graph of the secant function has vertical asymptotes at odd multiples of . Also notice that the secant function, like the cosine function, is periodic with fundamental period of 2π (or 360°).

15. Secant Graph

16. Cosecant Graph

17. Example • Find the value of each expression to four decimal places. • Tan 124° b. Cot 245° c. Csc 4 d. Sec 3

18. Example • Express each of the following in terms of a reference angle. • csc 300° b. sec 135 ° c. cot 315° d. tan (-135°)

19. Example • Find the exact value of each expression or state that the value is undefined. • Cot 420° b. Csc c. Sec d. Tan 225°

20. Example If csc Θ = and -90° < Θ < 90°, find the values of the other five trigonometric functions.