Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The angle
theta_(1) is located in Quadrant III, and
sin(theta_(1))=-(12)/(13).
What is the value of
cos(theta_(1)) ?
Express your answer exactly.

cos(theta_(1))=

◻

The angle $\theta_{1}$ is located in Quadrant III, and $\sin \left(\theta_{1}\right)=-\frac{12}{13}$ . $\newline$ What is the value of $\cos \left(\theta_{1}\right)$ ? $\newline$ Express your answer exactly. $\newline$ $\cos \left(\theta_{1}\right)=$ $\newline$ $\square$

View step-by-step help

View step-by-step help

Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q. The angle

\theta_{1}

is located in Quadrant III, and

\sin \left(\theta_{1}\right)=-\frac{12}{13}

.

\newline

What is the value of

\cos \left(\theta_{1}\right)

?

\newline

Express your answer exactly.

\newline

\cos \left(\theta_{1}\right)=

\newline

\square

Apply Pythagorean Identity: Use the Pythagorean identity for sine and cosine. $\newline$ The Pythagorean identity states that $\sin^2(\theta) + \cos^2(\theta) = 1$ . $\newline$ Since we know $\sin(\theta_{1}) = -\frac{12}{13}$ , we can substitute this into the identity to find $\cos(\theta_{1})$ .
Substitute $\sin(\theta_{1})$ : Substitute the value of $\sin(\theta_{1})$ into the Pythagorean identity. $\newline$ $\sin^2(\theta_{1}) + \cos^2(\theta_{1}) = 1$ $\newline$ $(-\frac{12}{13})^2 + \cos^2(\theta_{1}) = 1$ $\newline$ $\frac{144}{169} + \cos^2(\theta_{1}) = 1$
Solve for $\cos^2(\theta_{1})$ : Solve for $\cos^2(\theta_{1})$ .
$\cos^2(\theta_{1}) = 1 - \frac{144}{169}$
$\cos^2(\theta_{1}) = \frac{169}{169} - \frac{144}{169}$
$\cos^2(\theta_{1}) = \frac{169 - 144}{169}$
$\cos^2(\theta_{1}) = \frac{25}{169}$
Find $\cos(\theta_{1})$ : Take the square root of both sides to find $\cos(\theta_{1})$ . Since $\theta_{1}$ is in Quadrant III, where cosine is negative, we take the negative square root. $\cos(\theta_{1}) = -\sqrt{\frac{25}{169}}$ $\cos(\theta_{1}) = -\frac{5}{13}$

More problems from Find trigonometric ratios using a Pythagorean or reciprocal identity

Question

\cos(\theta)=\frac{8}{17}

0^\circ<\theta<90^\circ

\sec(\theta)

? Write your answer in simplified, rationalized form.

\sec(\theta)=

Posted 2 months ago

Question

\cos(x) = -0.3

\sin(90^\circ - x)

Posted 2 months ago

Question

\csc(\theta) = \frac{17}{8}

0^\circ < \theta < 90^\circ

\tan(\theta)

\newline

Write your answer in simplified, rationalized form.

\newline

\tan(\theta) =

\newline

Posted 2 months ago

Question

Is the sine function even or odd?

\newline

\newline

\text{[A]even}

\newline

\text{[B]odd}

Posted 2 months ago

Question

Write the sentence as an equation.

\newline

89

is equal to the quotient of

y

17

\newline

/

) if you want to use a division sign.

\newline

Posted 1 month ago

Question

Write the sentence as an equation.

\newline

45

is equal to the quotient of

z

5

\newline

/

) if you want to use a division sign.

\newline

Posted 3 months ago

Question

h

\newline

h^2 + 24h = 0

\newline

Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.

\newline

h =

Posted 3 months ago

Question

h^2 - 4h = 0

Posted 1 month ago

Question

Combine the like terms to create an equivalent expression:

\newline

-k-(-8k)=\square

Posted 4 months ago

Question

Which of the following degree measures is equal to

5 \pi

\newline

(The number of degrees of arc in a circle is

360

. The number of radians of arc in a circle is

2 \pi

\newline

1

\newline

144^{\circ}

\newline

900^{\circ}

\newline

1,080^{\circ}

\newline

1,800^{\circ}

Posted 1 month ago