Question

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Answer 1

The appropriate values are; $sin \theta = \frac{3\sqrt{10} }{10 }$, $cos\theta = -\frac{\sqrt{10} }{10}$, $cot\theta= -\frac{1}{3}$, $sec\theta=-\sqrt{10}$ and $csc\theta= \frac{\sqrt{10} }{3}$.

What are the trig values?

Given the equation and interval.

tanθ = -3, [ π/2 < θ < π ]

First, we use the definition of tangent to determine the known sides of the unit circle right triangle.

Note that; the quadrant determines the sign of each values.

tanθ = opposite / hypotenuse

We can use Pythagoras theorem to find the hypotenuse of the unit circle right triangle as the opposite and adjacent sides are known.

Hypotenuse = √( opposite² + adjacent² )

Hence, we have;

Hypotenuse = √( [3]² + [-1]² )

Hypotenuse = √( 9 + 1 )

Hypotenuse = √10

1) To find sinθ

sinθ = opposite / hypotenuse

sinθ = 3/√10

We simplify

sinθ = 3/√10 × √10/√10

sinθ = 3√10 / 10

$sin \theta = \frac{3\sqrt{10} }{10 }$

2) cosθ

cosθ = adjacent / hypotenuse

cosθ = -1 / √10

Simplify

cosθ = -1 / √10 × √10/√10

cosθ = -√10 / 10

$cos\theta = -\frac{\sqrt{10} }{10}$

3) cotθ

cotθ = adjacent / opposite

cotθ = -1 / 3

$cot\theta= -\frac{1}{3}$

4) secθ

secθ = hypotenuse / adjacent

secθ = √10 / -1

$sec\theta=-\sqrt{10}$

5) cscθ

cscθ = hypotenuse / opposite

cscθ = √10 / 3

$csc\theta= \frac{\sqrt{10} }{3}$

Therefore, the appropriate values are; $sin \theta = \frac{3\sqrt{10} }{10 }$, $cos\theta = -\frac{\sqrt{10} }{10}$, $cot\theta= -\frac{1}{3}$, $sec\theta=-\sqrt{10}$ and $csc\theta= \frac{\sqrt{10} }{3}$.

Learn more about trig ratios here: https://brainly.com/question/14977354

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