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Let's break down and solve the two parts of the problem:

### (i) Explain why the unlabeled angle is \( 90^\circ - \theta \).

In a right triangle, the sum of the angles is always \( 180^\circ \). One of the angles is \( 90^\circ \) (the right angle), and another angle is labeled \( \theta \). Therefore, the remaining angle must be:

\[
180^\circ - 90^\circ - \theta = 90^\circ - \theta
\]

Thus, the unlabeled angle is \( 90^\circ - \theta \) because the sum of the angles in any triangle must equal \( 180^\circ \).

### (ii) Complete the following trigonometric identities.

Now let's complete the identities using known trigonometric relationships:

#### First identity: 
\[
\sin \theta = \cos (90^\circ - \theta)
\]

This identity is derived from the co-function identity for sine and cosine. The sine of an angle is equal to the cosine of its complement.

#### Second identity: 
\[
\sin(90^\circ - \theta) = \cos \theta
\]

This is another co-function identity. It states that the sine of an angle’s complement is equal to the cosine of the angle.

#### Third identity: 
\[
\cos \theta = \sin (90^\circ - \theta)
\]

This is just the reverse of the first identity, stating that the cosine of an angle is equal to the sine of its complement.

#### Fourth identity: 
\[
\cos(90^\circ - \theta) = \sin \theta
\]

This is again the reverse of the second identity, confirming the co-function relationship between sine and cosine for complementary angles.

### Completed identities:
\[
\sin \theta = \cos(90^\circ - \theta)
\]
\[
\sin(90^\circ - \theta) = \cos \theta
\]
\[
\cos \theta = \sin(90^\circ - \theta)
\]
\[
\cos(90^\circ - \theta) = \sin \theta
\]

Would you like further details or explanations on any part of this problem?

---

### Related Questions:
1. What is the relationship between tangent and cotangent for complementary angles?
2. Can you explain why the sum of the angles in any triangle is \(180^\circ\)?
3. How do the sine and cosine functions behave for angles greater than \(90^\circ\)?
4. What are other trigonometric co-function identities apart from sine and cosine?
5. How can you derive the sine and cosine values for special angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\)?

### Tip:
Always remember that the sine of an angle is equal to the cosine of its complement, and vice versa. This property can be very useful in solving trigonometric problems.

Using the right triangle provided, explain why the unlabeled angle is 90° - θ and complete the given trigonometric identities.

Solve a problem involving the unlabeled angle in a right triangle and complete trigonometric identities. This exercise covers core concepts of trigonometry, including sine, cosine, and angle sums, suitable for Grades 9-12.

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