If \( \tan 35^{\circ}=m \), determine the value of the following in terms of \( m \) : 5.1.1 \( \sin 215^{\circ} \) 5.1.2 \( \sin 70^{\circ} \)

We are given that tan 35° = m. Our goal is to express sin 215° and sin 70° in terms of m. Step 1. Express sin 35° and cos 35° in terms of m. Since   tan 35° = m = sin 35°/cos 35°, we have   sin 35° = m · cos 35°. Using the Pythagorean identity:   sin² 35° + cos² 35° = 1, substitute sin 35° = m cos 35°:   (m cos 35°)² + cos² 35° = m² cos² 35° + cos² 35° = cos² 35° (m² + 1) = 1. Thus,   cos² 35° = 1/(m² + 1), and since 35° is in the first quadrant (where cosine is positive)   cos 35° = 1/√(m² + 1). Then,   sin 35° = m/√(m² + 1). Step 2. Find sin 215°. Notice that:   215° = 180° + 35°. The sine function in the third quadrant (angles between 180° and 270°) is negative, and we have:   sin(180° + θ) = - sin θ. So:   sin 215° = - sin 35° = - (m/√(m² + 1)). Step 3. Find sin 70°. There are a couple of ways to work this out. One convenient method is to use the double-angle formula applied to 35° since:   70° = 2 × 35°. Recall that:   sin 2θ = 2 sin θ cos θ. Thus:   sin 70° = 2 sin 35° cos 35°. Substitute the expressions found earlier:   sin 70° = 2 · (m/√(m² + 1)) · (1/√(m² + 1))        = 2m/(m² + 1). Final Answers: 5.1.1 sin 215° = – m/√(m² + 1) 5.1.2 sin 70° = 2m/(m² + 1)