https://file.aiquickdraw.com/mathbot/file/17277991566938uocsu2s.jpg

Let's work through the problem from the image.

We are given two right-angled triangles and asked to:

(a) Determine the value of:
   (i) \(\sin c\)
   (ii) \(\cos d\)

(b) Find the exact value of \(\sin (c + d)\).

### Step 1: Analyze the two triangles

#### First Triangle:
- One of the triangles has sides labeled \( \sqrt{7} \) and an unknown vertical side.
- This triangle contains angle \(c\).

To find \( \sin c \), we need:
\[
\sin c = \frac{\text{opposite}}{\text{hypotenuse}}
\]

However, we need the length of the hypotenuse. Let's calculate it using the Pythagorean theorem:
\[
\text{hypotenuse}^2 = (\sqrt{7})^2 + (\text{other side})^2 = 7 + \text{other side}^2
\]
We need more details to complete this.

Would you like me to proceed with detailed explanations of each step or clarify anything before we continue? 

Here are 5 questions to consider expanding on this:

1. How is the Pythagorean theorem applied to right triangles?
2. What is the relationship between sine and cosine in a right triangle?
3. Could the angles \(c\) and \(d\) be complementary?
4. What is the sum of angles in any triangle?
5. Can trigonometric identities help simplify complex expressions?

Tip: Always double-check which sides of the triangle correspond to the opposite, adjacent, and hypotenuse for the angle in question.

Two right-angled triangles are shown below. (a) Determine the value of: (i) sin(c), (ii) cos(d). (b) Find the exact value of sin(c + d).

This problem explores the trigonometric ratios sine and cosine in right-angled triangles. Learn how to determine sin(c), cos(d), and the exact value of sin(c + d) using trigonometric identities and the Pythagorean theorem. Suitable for Grades 9-11.

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