Question

Evaluate each of the following expressions without using a calculator. Use radian measures.

Answer 1

Answer:

e) $\pi/2$

f) $3\pi/2$

g) $0$

h) $\pi$

i) $\pi/4$

j) $3\pi/4$

Step-by-step explanation:

We can solve this exercise by watching a basic trigonometric table

Here i will list values of sin and cos in the points $0, \pi/4, \pi/2, 3\pi/4$  and $\pi$

• $cos(0) = 1$
• $sin(0) = 0$
• $cos(\pi/4) = \sqrt2 / 2$
• $sin(\pi/4) = \sqrt2 / 2$
• $cos(\pi/2) = 0$
• $sin(\pi/2) = 1$
• $cos(3 \pi/4) = -\sqrt2 / 2$
• $sin(3 \pi/4) = \sqrt2 / 2$
• $cos(\pi) = -1$
• $sin(\pi) = 0$

Hence, the sin takes the value 1 on $\pi/2$, while it takes the value -1 in its opposite value - \pi/2 (because it is an odd function). If we use the periodicity of the sin, then sin(- \pi/2 + 2 \pi) = -1. Hence $sin^{-1}(-1) = 3 \pi/2$

The cos takes value 1 in 0, and it takes the value -1 in $\pi$

The tangent is the quotient between sin and cos. The tangent is 1 when both cos and sin are equal. We can see that they are equal in $\pi/4$, where they take the value $\sqrt2 / 2$

The tangent is -1 where they are the opposite. This happens in $3 \pi/4$ where they take the value $\sqrt 2/2$ and $- \sqrt2 /2$ respectively.