Math / Trigonometry

Question $\begin{array}{l} \csc \theta=\frac{16}{5} \\ \sin \theta=\frac{5}{16} \end{array}$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) $\cos \theta=\square$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Studdy Solution

STEP 1

What is this asking? Given the cosecant of an angle $\theta$ is $\frac{16}{5}$ , we need to find the cosine of that same angle. Watch out! Remember the relationship between sine, cosine, and the Pythagorean identity!
Don't mix up the sides of your right triangle.

STEP 2

1. Find the sine of the angle.
2. Use the Pythagorean identity to find the cosine of the angle.

STEP 3

We're given that $\csc \theta = \frac{16}{5}$ .
Remember that cosecant is the reciprocal of sine!
So, if $\csc \theta = \frac{16}{5}$ , then $\sin \theta = \frac{1}{\csc \theta}$ .

STEP 4

Let's calculate the sine of $\theta$ : $\sin \theta = \frac{1}{\frac{16}{5}} = \frac{5}{16}$ So, $\sin \theta = \frac{\textbf{5}}{\textbf{16}}$ .
Awesome!

STEP 5

The Pythagorean identity states: $\sin^2 \theta + \cos^2 \theta = 1$ This is a superstar equation in trigonometry!
It connects sine and cosine.
We know $\sin \theta$ , so we can use this identity to find $\cos \theta$ .

STEP 6

Let's plug in our value for $\sin \theta$ : $\left(\frac{5}{16}\right)^2 + \cos^2 \theta = 1$ $\frac{25}{256} + \cos^2 \theta = 1$

STEP 7

Now, let's isolate $\cos^2 \theta$ by subtracting $\frac{25}{256}$ from both sides of the equation: $\cos^2 \theta = 1 - \frac{25}{256}$ To subtract, we need a common denominator, which is 256.
We can rewrite $1$ as $\frac{256}{256}$ : $\cos^2 \theta = \frac{256}{256} - \frac{25}{256} = \frac{\textbf{231}}{\textbf{256}}$

STEP 8

To find $\cos \theta$ , we take the square root of both sides: $\cos \theta = \pm \sqrt{\frac{231}{256}} = \pm \frac{\sqrt{231}}{\sqrt{256}} = \pm \frac{\sqrt{231}}{16}$ So, $\cos \theta = \pm \frac{\sqrt{\textbf{231}}}{\textbf{16}}$ .
We've got our cosine value!

STEP 9

$\cos \theta = \pm \frac{\sqrt{231}}{16}$

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Studdy Solution

STEP 1

What is this asking? Given the cosecant of an angle $\theta$ is $\frac{16}{5}$ , we need to find the cosine of that same angle. Watch out! Remember the relationship between sine, cosine, and the Pythagorean identity!
Don't mix up the sides of your right triangle.

STEP 2

1. Find the sine of the angle.
2. Use the Pythagorean identity to find the cosine of the angle.

STEP 3

We're given that $\csc \theta = \frac{16}{5}$ .
Remember that cosecant is the reciprocal of sine!
So, if $\csc \theta = \frac{16}{5}$ , then $\sin \theta = \frac{1}{\csc \theta}$ .

STEP 4

Let's calculate the sine of $\theta$ : $\sin \theta = \frac{1}{\frac{16}{5}} = \frac{5}{16}$ So, $\sin \theta = \frac{\textbf{5}}{\textbf{16}}$ .
Awesome!

STEP 5

The Pythagorean identity states: $\sin^2 \theta + \cos^2 \theta = 1$ This is a superstar equation in trigonometry!
It connects sine and cosine.
We know $\sin \theta$ , so we can use this identity to find $\cos \theta$ .

STEP 6

Let's plug in our value for $\sin \theta$ : $\left(\frac{5}{16}\right)^2 + \cos^2 \theta = 1$ $\frac{25}{256} + \cos^2 \theta = 1$

STEP 7

STEP 8

To find $\cos \theta$ , we take the square root of both sides: $\cos \theta = \pm \sqrt{\frac{231}{256}} = \pm \frac{\sqrt{231}}{\sqrt{256}} = \pm \frac{\sqrt{231}}{16}$ So, $\cos \theta = \pm \frac{\sqrt{\textbf{231}}}{\textbf{16}}$ .
We've got our cosine value!

STEP 9

$\cos \theta = \pm \frac{\sqrt{231}}{16}$

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Studdy Solution

STEP 1

What is this asking? Given the cosecant of an angle θ\thetaθ is 165\frac{16}{5}516​, we need to find the cosine of that same angle. Watch out! Remember the relationship between sine, cosine, and the Pythagorean identity!Don't mix up the sides of your right triangle.

STEP 2

1. Find the sine of the angle.2. Use the Pythagorean identity to find the cosine of the angle.

STEP 3

We're given that csc⁡θ=165\csc \theta = \frac{16}{5}cscθ=516​.Remember that cosecant is the reciprocal of sine!So, if csc⁡θ=165\csc \theta = \frac{16}{5}cscθ=516​, then sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}sinθ=cscθ1​.

STEP 4

Let's **calculate** the sine of θ\thetaθ: sin⁡θ=1165=516 \sin \theta = \frac{1}{\frac{16}{5}} = \frac{5}{16} sinθ=516​1​=165​ So, sin⁡θ=516\sin \theta = \frac{\textbf{5}}{\textbf{16}}sinθ=165​.Awesome!

STEP 5

The **Pythagorean identity** states: sin⁡2θ+cos⁡2θ=1 \sin^2 \theta + \cos^2 \theta = 1 sin2θ+cos2θ=1 This is a superstar equation in trigonometry!It connects sine and cosine.We know sin⁡θ\sin \thetasinθ, so we can use this identity to find cos⁡θ\cos \thetacosθ.

STEP 6

Let's **plug in** our value for sin⁡θ\sin \thetasinθ: (516)2+cos⁡2θ=1 \left(\frac{5}{16}\right)^2 + \cos^2 \theta = 1 (165​)2+cos2θ=1 25256+cos⁡2θ=1 \frac{25}{256} + \cos^2 \theta = 1 25625​+cos2θ=1

STEP 7

STEP 8

STEP 9

cos⁡θ=±23116\cos \theta = \pm \frac{\sqrt{231}}{16}cosθ=±16231​​

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? Given the cosecant of an angle $\theta$ is $\frac{16}{5}$ , we need to find the cosine of that same angle. Watch out! Remember the relationship between sine, cosine, and the Pythagorean identity!
Don't mix up the sides of your right triangle.

1. Find the sine of the angle.
2. Use the Pythagorean identity to find the cosine of the angle.

We're given that $\csc \theta = \frac{16}{5}$ .
Remember that cosecant is the reciprocal of sine!
So, if $\csc \theta = \frac{16}{5}$ , then $\sin \theta = \frac{1}{\csc \theta}$ .

Let's calculate the sine of $\theta$ : $\sin \theta = \frac{1}{\frac{16}{5}} = \frac{5}{16}$ So, $\sin \theta = \frac{\textbf{5}}{\textbf{16}}$ .
Awesome!

The Pythagorean identity states: $\sin^2 \theta + \cos^2 \theta = 1$ This is a superstar equation in trigonometry!
It connects sine and cosine.
We know $\sin \theta$ , so we can use this identity to find $\cos \theta$ .

Let's plug in our value for $\sin \theta$ : $\left(\frac{5}{16}\right)^2 + \cos^2 \theta = 1$ $\frac{25}{256} + \cos^2 \theta = 1$

$\cos \theta = \pm \frac{\sqrt{231}}{16}$