𝒔𝒆𝒄²𝜽 + 𝒕𝒂𝒏²𝜽 = 𝟕 find the value of Sin𝜽 =?

We are given the equation:

\[
\sec^2\theta + \tan^2\theta = 7
\]

### Step 1: Using a trigonometric identity
We know from trigonometric identities that:

\[
\sec^2\theta = 1 + \tan^2\theta
\]

Substitute \(\sec^2\theta\) into the original equation:

\[
(1 + \tan^2\theta) + \tan^2\theta = 7
\]

Simplifying:

\[
1 + 2\tan^2\theta = 7
\]

\[
2\tan^2\theta = 6
\]

\[
\tan^2\theta = 3
\]

### Step 2: Solving for \(\tan\theta\)

Taking the square root of both sides:

\[
\tan\theta = \pm\sqrt{3}
\]

### Step 3: Using another identity to find \(\sin\theta\)
We know the identity:

\[
\sin^2\theta = \frac{\tan^2\theta}{1 + \tan^2\theta}
\]

Substituting \(\tan^2\theta = 3\):

\[
\sin^2\theta = \frac{3}{1 + 3} = \frac{3}{4}
\]

Taking the square root:

\[
\sin\theta = \pm\frac{\sqrt{3}}{2}
\]

Therefore, the value of \(\sin\theta\) is either \(\frac{\sqrt{3}}{2}\) or \(-\frac{\sqrt{3}}{2}\), depending on the quadrant of \(\theta\).

Would you like more details on how to determine the quadrant? 

Here are 5 related questions:
1. What are the values of trigonometric functions in different quadrants?
2. How is the Pythagorean identity used in solving trigonometric equations?
3. What is the relationship between \(\sec\theta\) and \(\cos\theta\)?
4. How do you determine the sign of \(\sin\theta\) based on \(\tan\theta\)?
5. How can you apply the tangent identity in solving other trigonometric problems?

**Tip**: Always consider the quadrant to determine the correct sign of trigonometric values.

sec²θ + tan²θ = 7, find the value of Sinθ?

Learn how to solve the trigonometric equation sec²θ + tan²θ = 7 and find the value of sinθ using Pythagorean identities. This step-by-step solution covers key trigonometric concepts suitable for students in Grades 10-12.

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