1. **Problem Statement:** Find the missing angle \(\angle C\) in a right triangle using the cosine ratio. 2. **Recall the cosine ratio:** \[ \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \] This means the cosine of an angle in a right triangle equals the length of the side adjacent to the angle divided by the hypotenuse. 3. **Example 4:** Given triangle ABC with \(\angle C = 90^\circ\), side AC = 18, and hypotenuse AB = 23, find \(\angle C\). 4. Since \(\angle C = 90^\circ\), the missing angle to find is either \(\angle A\) or \(\angle B\). Usually, we find \(\angle A\) or \(\angle B\) using cosine. 5. Let's find \(\angle A\) using the cosine ratio: \[ \cos(\angle A) = \frac{\text{adjacent side to } A}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{18}{23} \] 6. Calculate \(\cos(\angle A)\): \[ \cos(\angle A) \approx 0.7826 \] 7. Find \(\angle A\) by taking the inverse cosine: \[ \angle A = \cos^{-1}(0.7826) \approx 38.68^\circ \] 8. Since the triangle's angles sum to 180°, and \(\angle C = 90^\circ\), find \(\angle B\): \[ \angle B = 90^\circ - \angle A = 90^\circ - 38.68^\circ = 51.32^\circ \] 9. **Answer:** The missing angle \(\angle C\) is the right angle (90°), and the other angles are approximately \(38.68^\circ\) and \(51.32^\circ\). --- For the other triangles: **Triangle c:** Given \(\angle B = 48^\circ\), \(\angle C = 90^\circ\), find \(x = AC\). Use the cosine ratio for \(\angle B\): \[ \cos(48^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{4}{x} \] Solve for \(x\): \[ x = \frac{4}{\cos(48^\circ)} \approx \frac{4}{0.6691} \approx 5.98 \text{ m} \] **Triangle d:** Given \(\angle K = 62^\circ\), \(\angle M = 90^\circ\), side KM = 13.4 cm, find \(x = LM\). Use the cosine ratio for \(\angle K\): \[ \cos(62^\circ) = \frac{13.4}{x} \] Solve for \(x\): \[ x = \frac{13.4}{\cos(62^\circ)} \approx \frac{13.4}{0.4695} \approx 28.54 \text{ cm} \] **Summary:** - Triangle c: \(x \approx 5.98\) m - Triangle d: \(x \approx 28.54\) cm - Example 4: \(\angle A \approx 38.68^\circ\), \(\angle B \approx 51.32^\circ\), \(\angle C = 90^\circ\)