Question

Question 3 (Essay Worth 4 points)

Find the exact value by using a half-angle identity.

tan (7π/8)

Answer 1

`\displaystyle \tan \left((7\pi)/(8)\right) = -\left(2 + √(2)\right)`.

Explanation:

Assume that the following are known:

• The tangent half-angle identity:
  `\displaystyle \tan\left((\theta)/(2)\right) = (1 - \cos(\theta))/(\sin(\theta))`.
• The sine and cosine of
  `\displaystyle (\pi)/(4)`:
  `\displaystyle \sin\left(\displaystyle (\pi)/(4)\right) = (√(2))/(2)`,
  `\displaystyle \cos\left(\displaystyle (\pi)/(4)\right) = (√(2))/(2)`.
• The trigonometric identities for the sine and cosine of supplementary angles:
  `\sin(\theta) = -\sin(2\pi - \theta)` and
  `\cos(\theta) = \cos(2\pi - \theta)`.

The angle in question
`\displaystyle (7 \pi)/(8)` isn't very well-known. However, note that:

• 
  `\displaystyle (7 \pi)/(8)` is equal to one-half times the angle
  `\displaystyle (7\pi)/(4)`;
• 
  `\displaystyle (7\pi)/(4)` is equal to
  `2\pi` minus
  `\displaystyle (\pi)/(4)`, which is an angle with known sine and cosine values.

Since the angle
`\displaystyle (7\pi)/(4)` is equal to
`\displaystyle 2\pi - (\pi)/(4)`, its sine and cosine can be found from the sine and cosine of
`\displaystyle (\pi)/(4)` using the trigonometric identities
`\sin(2\pi - \theta) = -\sin(\theta)` and
`\cos(2\pi - \theta) = \cos(\theta)`:

• 
  `\displaystyle \sin\left((7\pi)/(4)\right) = -\sin\left(2\pi - (7\pi)/(4)\right) = - \sin\left((\pi)/(4)\right) = - (√(2))/(2)`.
• 
  `\displaystyle \cos\left((7\pi)/(4)\right) = \cos\left(2\pi - (7\pi)/(4)\right) = \cos\left((\pi)/(4)\right) = (√(2))/(2)`.

Apply the tangent half-angle identity to find the tangent of
`\displaystyle (7 \pi)/(8)`:

`\begin{aligned}\tan\left((7\pi)/(8)\right) &= \tan\left((7\pi/4)/(2)\right) = (1 - \cos(7\pi/4))/(\sin(7\pi/4)) \\ &= (1 - \left(√(2)/2\right))/(√(2) / 2) = (2 - √(2))/(√(2)) = (\left(√(2)\right)^2 - √(2))/(√(2)) = √(2) - 1 \end{aligned}`.