Question

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

e. sin−1(1) f. sin−1(−1)
g. cos−1(1) h. cos−1(−1)
i. tan−1(1) j. tan−1(−1)

Answer 1

e)
`\pi/2`

f)
`3\pi/2`

g)
`0`

h)
`\pi`

i)
`\pi/4`

j)
`3\pi/4`

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.