Question

Find the exact value of tan 5pi / 4. No decimals are allowed, no calculators are allowed. Show your work.

Part 1: Sketch the angle. Identify the reference angle on your sketch.
Part 2: Identify the terminal point associated with the angle, then find the tangent of the angle.

Answer 1

Explanation:

Most people are more familiar with degrees than with radians. So perhaps what is giving a bad time is the pi/4

pi/4 = 45 degrees.

Tangent of 45 degrees is 1. So part A is = 5 * 45 degrees = 225 degrees which is in the 3 rd quadrant. (Remember that the 4 goes into defining the 45 degree angle from pi/4 radians.)

The tangent is positive in quad 3. So you draw a line 1/2 way between the minus x and minus y axis to get 5/4 pi.

The tangent of the angle is 1. The terminal point is (-1 , - 1)

Answer 2

The exact value of
`\tan ((5\pi)/(4))` is 1.

The reference angle is
`(\pi)/(4)`

The terminal point is
`(-(1)/(√(2)),-(1)/(√(2)))`.

Explanation:

The given expression is

`\tan ((5\pi)/(4))`

Since
`\pi<(5\pi)/(4)<(3\pi)/(2)`, so it lies in third quadrant.

We need to find the exact value of given expression.

`\tan ((5\pi)/(4))=\tan (pi+(\pi)/(4))`

In third quadrant, tangent is positive.

`\tan ((5\pi)/(4))=\tan (pi+(\pi)/(4))`

`\tan ((5\pi)/(4))=\tan ((\pi)/(4))`
`[\because \tan (\pi+\theta)=\tan \theta]`

`\tan ((5\pi)/(4))=1`
`[\because \tan ((\pi)/(4))=1]`

Part 1: Reference angle for third quadrant is

`\text{Reference angle}=\theta -\pi`

`\text{Reference angle}=(5\pi)/(4) -\pi`

`\text{Reference angle}=(\pi)/(4)`

Therefore the reference angle is
`(\pi)/(4)`.

Part 2: Terminal point associated with the angle be (x,y).

For a unit circle hypotenuse is 1, perpendicular is y and base is x.

`\sin (\theta)=(perpendicular)/(hypotenuse)`

`\sin ((\pi)/(4))=(y)/(1)`

`(1)/(√(2))=y`

`\cos (\theta)=(base)/(hypotenuse)`

`\cos ((\pi)/(4))=(x)/(1)`

`(1)/(√(2))=x`

In third quadrant sine and cosine both are negative. So the terminal point is
`(-(1)/(√(2)),-(1)/(√(2)))`.