If

sin
(
θ
)
=
7
8
and
θ
is in the 1st quadrant, find
cos
(
θ
)
.

Given that \(\sin(\theta) = \frac{7}{8}\) and \(\theta\) is in the 1st quadrant, we can find \(\cos(\theta)\) using the Pythagorean identity:

\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

### Step 1: Substitute the given value of \(\sin(\theta)\) into the identity.

\[
\left(\frac{7}{8}\right)^2 + \cos^2(\theta) = 1
\]

\[
\frac{49}{64} + \cos^2(\theta) = 1
\]

### Step 2: Solve for \(\cos^2(\theta)\).

\[
\cos^2(\theta) = 1 - \frac{49}{64}
\]

\[
\cos^2(\theta) = \frac{64}{64} - \frac{49}{64} = \frac{15}{64}
\]

### Step 3: Take the square root of both sides.

Since \(\theta\) is in the 1st quadrant, \(\cos(\theta)\) is positive:

\[
\cos(\theta) = \sqrt{\frac{15}{64}} = \frac{\sqrt{15}}{8}
\]

So, \(\cos(\theta) = \frac{\sqrt{15}}{8}\).

Would you like more details, or do you have any questions?

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Here are 5 related questions for further practice:

1. If \(\sin(\theta) = \frac{7}{8}\) and \(\theta\) is in the 2nd quadrant, what is \(\cos(\theta)\)?
2. Find \(\tan(\theta)\) given \(\sin(\theta) = \frac{7}{8}\) in the 1st quadrant.
3. If \(\sin(\theta) = \frac{7}{8}\), find \(\csc(\theta)\).
4. Determine the value of \(\theta\) if \(\sin(\theta) = \frac{7}{8}\) and \(\theta\) is in the 1st quadrant.
5. If \(\cos(\theta) = \frac{3}{5}\), find \(\sin(\theta)\) when \(\theta\) is in the 1st quadrant.

**Tip:** In trigonometry, always consider the quadrant when determining the sign of trigonometric functions.

If sin(θ) = 7/8 and θ is in the 1st quadrant, find cos(θ).

Learn how to find cos(θ) when sin(θ) = 7/8 and θ is in the 1st quadrant. This solution uses the Pythagorean identity to calculate the value of cos(θ), including step-by-step details. Ideal for students in Grades 9-11.

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