Question

EXERCISE 8 (a) If \( \tan 35^{\circ}=k \), write the following in terms of \( k \) : (1) \( \tan 145^{\circ} \) (2) \( \tan 215^{\circ} \) (3) \( \tan \left(-325^{\circ}\right) \) (4) \( \sin 35^{\circ} \) (5) \( \cos 325^{\circ} \)

Ask by Ross Mathis. in South Africa
Mar 07,2025

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1. \( \tan 145^\circ = -k \) 2. \( \tan 215^\circ = k \) 3. \( \tan (-325^\circ) = k \) 4. \( \sin 35^\circ = \frac{k}{\sqrt{1+k^2}} \) 5. \( \cos 325^\circ = \frac{1}{\sqrt{1+k^2}} \)

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Bonus Knowledge

To express those trigonometric values in terms of \( k \), let's break them down: 1. To find \( \tan 145^{\circ} \): \[ \tan 145^{\circ} = \tan(180^{\circ} - 35^{\circ}) = -\tan 35^{\circ} = -k \] 2. To find \( \tan 215^{\circ} \): \[ \tan 215^{\circ} = \tan(180^{\circ} + 35^{\circ}) = \tan 35^{\circ} = k \] 3. To find \( \tan(-325^{\circ}) \): \[ -325^{\circ} = 360^{\circ} - 325^{\circ} = 35^{\circ} \\ \tan(-325^{\circ}) = \tan 35^{\circ} = k \] 4. For \( \sin 35^{\circ} \): Using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we have: \[ k = \frac{\sin 35^{\circ}}{\cos 35^{\circ}} \implies \sin 35^{\circ} = k \cdot \cos 35^{\circ} \] 5. For \( \cos 325^{\circ} \): \[ \cos 325^{\circ} = \cos(360^{\circ} - 35^{\circ}) = \cos 35^{\circ} \] In summary, the answers are: 1. \( \tan 145^{\circ} = -k \) 2. \( \tan 215^{\circ} = k \) 3. \( \tan(-325^{\circ}) = k \) 4. \( \sin 35^{\circ} = k \cdot \cos 35^{\circ} \) 5. \( \cos 325^{\circ} = \cos 35^{\circ} \)

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