Question

et theta equals negative 8 times pi over 3 period Part A: What is a coterminal angle of θ such that 0 ≤ θ ≤ 2π? (5 points)Part B: What are the exact values of all six trigonometric functions evaluated at θ? (10 points))

Answer 1

Therefore, the exact values of the six trigonometric functions at
`\(\theta = -(8\pi)/(3)\)` are:

`\[\sin \theta = -(√(3))/(2), \cos \theta = -(1)/(2), \tan \theta = -√(3), \csc \theta = -(2)/(√(3)), \sec \theta = -2, \cot \theta = -(1)/(√(3))\]`

Given:
`\(\theta = -(8\pi)/(3)\)`

Part A: Coterminal Angle of
`\(\theta\)` such that
`\(0 \leq \theta \leq 2\pi\)`

To find the coterminal angle within the range \(0 \leq \theta \leq 2\pi\), let's add
`\(2\pi\)` to
`\(\theta\)` until we obtain an angle within the desired range:

`\(\theta = -(8\pi)/(3)\)`

Adding
`\(2\pi\) to \(\theta\)`:

`\[\theta_{\text{co-terminal}} = -(8\pi)/(3) + 2\pi = -(8\pi)/(3) + (6\pi)/(3) = -(2\pi)/(3)\]`

Therefore, a coterminal angle of
`\(\theta\)` such that
`\(0 \leq \theta \leq 2\pi\) is \(-(2\pi)/(3)\).`

Part B: Exact Values of Trigonometric Functions at
`\(\theta\)`

To find the exact values of the trigonometric functions at
`\(\theta = -(8\pi)/(3)\)`, we can use the properties of trigonometric functions related to the unit circle.

Given
`\(\theta = -(8\pi)/(3)\)`, which is in the third quadrant (since it is more than
`(2\pi\))`:

Let's evaluate the trigonometric functions at this angle:

Given:

- Sine
`(\(\sin\)): \(\sin \theta = \sin\left(-(8\pi)/(3)\right)\)`

- Cosine
`(\(\cos\)): \(\cos \theta = \cos\left(-(8\pi)/(3)\right)\)`

- Tangent
`(\(\tan\)): \(\tan \theta = \tan\left(-(8\pi)/(3)\right)\)`

- Cosecant
`(\(\csc\)): \(\csc \theta = \csc\left(-(8\pi)/(3)\right)\)`

- Secant
`(\(\sec\)): \(\sec \theta = \sec\left(-(8\pi)/(3)\right)\)`

- Cotangent
`(\(\cot\)): \(\cot \theta = \cot\left(-(8\pi)/(3)\right)\)`

Let's calculate these values using the properties of trigonometric functions and the unit circle:

The reference angle for
`\((8\pi)/(3)\) is \((\pi)/(3)\).` In the third quadrant, sine and cosine are negative:

`\[\sin \theta = -\sin\left((\pi)/(3)\right) = -(√(3))/(2)\]`

`\[\cos \theta = -\cos\left((\pi)/(3)\right) = -(1)/(2)\]`

`\[\tan \theta = \tan\left((\pi)/(3)\right) = -√(3)\]`

`\[\csc \theta = -(1)/(\sin \theta) = -(2)/(√(3))\]`

`\[\sec \theta = -(1)/(\cos \theta) = -2\]`

`\[\cot \theta = -(1)/(\tan \theta) = -(1)/(√(3))\]`

Answer 2

Given

`\theta=-(8\pi)/(3)`

To find:

Part A: What is a coterminal angle of θ such that 0 ≤ θ ≤ 2π?

Part B: What are the exact values of all six trigonometric functions evaluated at θ?

Step-by-step explanation:

It is given that,

`\theta=-(8\pi)/(3)`

That implies,

Part A: