Find the exact value of the indicated trigonometric function, using the given information.
if ; terminal side of in Quadrant ।

Ask by Wade O'Quinn. in the United States

Jan 24,2025

Question

Find the exact value of the indicated trigonometric function, using the given information.
if ; terminal side of in Quadrant ।

Ask by Wade O'Quinn. in the United States

Jan 24,2025

UpStudy AI · Accepted Answer

To find the value of $$ \sin \theta $$ given that $$ \cos \theta = \frac{3}{5} $$ and that the terminal side of $$ \theta $$ is in Quadrant I, we can use the Pythagorean identity:

$$
\sin^2 \theta + \cos^2 \theta = 1
$$

### Step 1: Substitute the known value of $$ \cos \theta $$

We know that $$ \cos \theta = \frac{3}{5} $$. Therefore, we can substitute this value into the Pythagorean identity:

$$
\sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1
$$

### Step 2: Calculate $$ \left(\frac{3}{5}\right)^2 $$

Calculating $$ \left(\frac{3}{5}\right)^2 $$:

$$
\left(\frac{3}{5}\right)^2 = \frac{9}{25}
$$

### Step 3: Substitute and solve for $$ \sin^2 \theta $$

Now we substitute this back into the equation:

$$
\sin^2 \theta + \frac{9}{25} = 1
$$

To isolate $$ \sin^2 \theta $$, we subtract $$ \frac{9}{25} $$ from both sides:

$$
\sin^2 \theta = 1 - \frac{9}{25}
$$

### Step 4: Convert 1 to a fraction with a denominator of 25

We can express 1 as $$ \frac{25}{25} $$:

$$
\sin^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
$$

### Step 5: Take the square root

Now, we take the square root of both sides to find $$ \sin \theta $$:

$$
\sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5}
$$

### Step 6: Determine the sign of $$ \sin \theta $$

Since $$ \theta $$ is in Quadrant I, where sine is positive, we have:

$$
\sin \theta = \frac{4}{5}
$$

Thus, the exact value of $$ \sin \theta $$ is:

$$
\sin \theta = \frac{4}{5}
$$
$$ \sin \theta = \frac{4}{5} $$

Find the exact value of the indicated trigonometric function, using the given information.
if ; terminal side of in Quadrant ।

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Find the exact value of the indicated trigonometric function, using the given information. if ; terminal side of in Quadrant ।

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Find the exact value of the indicated trigonometric function, using the given information.
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