Question

Find cot and cos
If sec = -3 and sin 0 > 0

Answer 1

Second answer

Explanation:

We are given
`\displaystyle \large{\sec \theta = -3}` and
`\displaystyle \large{\sin \theta > 0}`. What we have to find are
`\displaystyle \large{\cot \theta}` and
`\displaystyle \large{\cos \theta}`.

First, convert
`\displaystyle \large{\sec \theta}` to
`\displaystyle \large{(1)/(\cos \theta)}` via trigonometric identity. That gives us a new equation in form of
`\displaystyle \large{\cos \theta}`:

`\displaystyle \large{(1)/(\cos \theta) = -3}`

Multiply
`\displaystyle \large{\cos \theta}` both sides to get rid of the denominator.

`\displaystyle \large{(1)/(\cos \theta) \cdot \cos \theta = -3 \cos \theta}\\\displaystyle \large{1=-3 \cos \theta}`

Then divide both sides by -3 to get
`\displaystyle \large{\cos \theta}`.

Hence,
`\displaystyle \large{\boxed{\cos \theta = - (1)/(3)}}`

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Next, to find
`\displaystyle \large{\cot \theta}`, convert it to
`\displaystyle \large{(1)/(\tan \theta)}` via trigonometric identity. Then we have to convert
`\displaystyle \large{\tan \theta}` to
`\displaystyle \large{(\sin \theta)/(\cos \theta)}` via another trigonometric identity. That gives us:

`\displaystyle \large{(1)/((\sin \theta)/(\cos \theta))}\\\displaystyle \large{(\cos \theta)/(\sin \theta)`

It seems that we do not know what
`\displaystyle \large{\sin \theta}` is but we can find it by using the identity
`\displaystyle \large{\sin \theta = √(1-\cos ^2 \theta)}` for
`\displaystyle \large{\sin \theta > 0}`.

From
`\displaystyle \large{\cos \theta = -(1)/(3)}` then
`\displaystyle \large{\cos ^2 \theta = (1)/(9)}`.

Therefore:

`\displaystyle \large{\sin \theta=\sqrt{1-(1)/(9)}}\\\displaystyle \large{\sin \theta = \sqrt{(9)/(9)-(1)/(9)}}\\\displaystyle \large{\sin \theta = \sqrt{(8)/(9)}}`

Then use the surd property to evaluate the square root.

Hence,
`\displaystyle \large{\boxed{\sin \theta=(2√(2))/(3)}}`

Now that we know what
`\displaystyle \large{\sin \theta}` is. We can evaluate
`\displaystyle \large{(\cos \theta)/(\sin \theta)}` which is another form or identity of
`\displaystyle \large{\cot \theta}`.

From the boxed values of
`\displaystyle \large{\cos \theta}` and
`\displaystyle \large{\sin \theta}`:-

`\displaystyle \large{\cot \theta = (\cos \theta)/(\sin \theta)}\\\displaystyle \large{\cot \theta = (-(1)/(3))/((2√(2))/(3))}\\\displaystyle \large{\cot \theta=-(1)/(3) \cdot (3)/(2√(2))}\\\displaystyle \large{\cot \theta=-(1)/(2√(2))`

Then rationalize the value by multiplying both numerator and denominator with the denominator.

`\displaystyle \large{\cot \theta = -(1 \cdot 2√(2))/(2√(2) \cdot 2√(2))}\\\displaystyle \large{\cot \theta = -(2√(2))/(8)}\\\displaystyle \large{\cot \theta = -(√(2))/(4)}`

Hence,
`\displaystyle \large{\boxed{\cot \theta = -(√(2))/(4)}}`

Therefore, the second choice is the answer.

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Summary

• Trigonometric Identity

`\displaystyle \large{\sec \theta = (1)/(\cos \theta)}\\ \displaystyle \large{\cot \theta = (1)/(\tan \theta) = (\cos \theta)/(\sin \theta)}\\ \displaystyle \large{\sin \theta = √(1-\cos ^2 \theta) \ \ \ (\sin \theta > 0)}\\ \displaystyle \large{\tan \theta = (\sin \theta)/(\cos \theta)}`

• Surd Property

`\displaystyle \large{\sqrt{(x)/(y)} = (√(x))/(√(y))}`

Let me know in the comment if you have any questions regarding this question or for clarification! Hope this helps as well.