QUESTION 5 If \( \tan 58^{\circ}=41 \), determine the following in tems of \( n \) withoul using a calculate \( \begin{array}{ll}\text { S.1.1 } & \sin 58^{\circ} \\ 5.1 .2 & \sin 296^{\circ} \\ 5.1 .3 & \cos 2^{\circ}\end{array} \)

To solve the problem, we will use trigonometric identities and properties of angles. Let's break it down step by step. ### Step 1: Find \( \sin 58^{\circ} \) We know that: \[ \tan 58^{\circ} = \frac{\sin 58^{\circ}}{\cos 58^{\circ}} \] Given that \( \tan 58^{\circ} = 41 \), we can express this as: \[ \frac{\sin 58^{\circ}}{\cos 58^{\circ}} = 41 \] This implies: \[ \sin 58^{\circ} = 41 \cos 58^{\circ} \] To find \( \sin 58^{\circ} \) in terms of \( n \), we can use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting \( \sin 58^{\circ} = 41 \cos 58^{\circ} \) into the identity: \[ (41 \cos 58^{\circ})^2 + \cos^2 58^{\circ} = 1 \] This simplifies to: \[ 1681 \cos^2 58^{\circ} + \cos^2 58^{\circ} = 1 \] \[ 1682 \cos^2 58^{\circ} = 1 \] \[ \cos^2 58^{\circ} = \frac{1}{1682} \] Thus, \[ \cos 58^{\circ} = \frac{1}{\sqrt{1682}} \] Now substituting back to find \( \sin 58^{\circ} \): \[ \sin 58^{\circ} = 41 \cos 58^{\circ} = 41 \cdot \frac{1}{\sqrt{1682}} = \frac{41}{\sqrt{1682}} \] ### Step 2: Find \( \sin 296^{\circ} \) Using the property of sine: \[ \sin(360^{\circ} - \theta) = -\sin \theta \] We have: \[ \sin 296^{\circ} = \sin(360^{\circ} - 64^{\circ}) = -\sin 64^{\circ} \] Using the co-function identity: \[ \sin(90^{\circ} - \theta) = \cos \theta \] Thus: \[ \sin 64^{\circ} = \cos 26^{\circ} \] So: \[ \sin 296^{\circ} = -\cos 26^{\circ} \] ### Step 3: Find \( \cos 2^{\circ} \) Using the cosine double angle identity: \[ \cos 2\theta = 2\cos^2 \theta - 1 \] We can express \( \cos 2^{\circ} \) as: \[ \cos 2^{\circ} = 2\cos^2 1^{\circ} - 1 \] However, we need to express this in terms of \( n \). Since we don't have a direct relationship, we can leave it as: \[ \cos 2^{\circ} = 2\cos^2 1^{\circ} - 1 \] ### Summary of Results 1. \( \sin 58^{\circ} = \frac{41}{\sqrt{1682}} \) 2. \( \sin 296^{\circ} = -\cos 26^{\circ} \) 3. \( \cos 2^{\circ} = 2\cos^2 1^{\circ} - 1 \) These results are expressed in terms of \( n \) as required.