2.4. EXERCISES 65
(a) tanx(b) cotx(c) secx
(d) cscx
(e) cosx
10. Solve the equations and give all solutions.
(a) sin(3x) =12
(b) cos(5x) =
√3
2(c) tan(x) =
√3
(d) sec(x) = 2
(e) sin(x+7) =
√2
2
(f) cos2 (x) =12
(g) sin4 (x) = 4
11. Sketch a graph of y = sinx.
12. Sketch a graph of y = cosx.
13. Sketch a graph of y = sin2x.
14. Sketch a graph of y = tanx.
15. Find a formula for sinxcosy in terms of sines and cosines of x+ y and x− y.
16. Using Problem 2 graph y = cos2 x.
17. If f (x) = Acosαx+Bsinαx, show there exists φ such that
f (x) =√
A2 +B2 sin(αx+φ) .
Show there also exists ψ such that f (x) =√
A2 +b2 cos(αx−ψ) . This is a veryimportant result, enough that some of these quantities are given names.
√A2 +B2 is
called the amplitude and φ or ψ are called phase shifts.
18. Using Problem 17 graph y = sinx+√
3cosx.
19. Give all solutions to sinx+√
3cosx =√
3. Hint: Use Problem 18.
20. As noted above 45o is the same angle as π/4 radians. Explain why 90o is the sameangle as π/2 radians. Next find a simple formula which will change the degreemeasure of an angle to radian measure and radian measure into degree measure.
21. Find a formula for tan(θ +β ) in terms of tanθ and tanβ
22. Find a formula for tan(2θ) in terms of tanθ .
23. Find a formula for tan(
θ
2
)in terms of tanθ .
24. Show tan(4θ) =4tanθ −4tan3 θ
1−6tan2 θ + tan4 θ. Use to show that
π
4= 4arctan
(15
)− arctan
(1
239
).