addition formulas for sin, cos, and tan and implications of them

Exercise Set 2.8

Figure 2.8.1. Relationships among
sine, cosine, and angles that differ by $\frac{\pi}{2}$.

1. Examine the diagrams in Figure 2.8.1. Use the Angle-Side-Angle theorem from geometry to prove that the two triangles are congruent, and therefore that $\cos(θ) = \sin(θ + \frac{\pi}{2})$ and that $\sin(θ) = -\cos(θ+\frac{\pi}{2})$.

2. Now that you have proved the statements in Exercise 2.7.1, use the animation in Figure 2.8.2 to convince yourself that they are always true. Pause the movie occasionally and examine the relationships shown in Figure 2.8.1. In what way does the diagram give compelling visual confirmation that $\cos(θ) = \sin(θ + \frac{\pi}{2})$ and that $\sin(θ) = -\cos(θ+\frac{\pi}{2})$ for all values of θ?

Figure 2.8.2. Relationships among sine, cosine, and angles that differ
by $\frac{\pi}{2}$.

3. The functions secant, cosecant, and cotangent are defined as $\sec(x) = \frac{1}{\cos(x)}$, $\csc(x) = \frac{1}{\sin(x)}$, and $\cot(x) = \frac{\cos(x)}{\sin(x)}$. Use the fact that $\cos(x) = \sin(x + \frac{\pi}{2})$ to express each of sec(x), csc(x), cot(
x), and tan(x) completely in terms of sin(x). This demonstrates that sin(x) is the only elementary trigonometric function we need, and that all other trigonometric functions can be defined in terms of it.

4. In the unit circle, $(\sin x)^2 + (\cos x)^2 = 1$ by the Pythagorean Theorem.

a) Use this fact to derive an equivalent identity involving tan(x).

b) Use this fact to derive an equivalent identity involving cot(x).

c) Use this fact to derive an equivalent identity involving csc(x).

d) Use this fact to derive an equivalent identity involving sec(x).