Trigonometry & Inverse Trigonometry
Angular Measurement and Comparison
Grade 11
Question:
<p>Which of the following is/are true?</p>
<p>(a) <span class="math">\sin 1 > \sin 1°</span></p>
<p>(b) <span class="math">\tan 1 > \tan 1°</span></p>
<p>(c) <span class="math">\sin 4 > \sin 4°</span></p>
<p>(d) <span class="math">\tan 4 > \tan 4°</span></p>
Step-by-Step Solution
Key Concept: Distinguish between radian and degree measures, noting that values around π/2 ≈ 1.57 radians behave differently from those near 4 radians.
<p>(a) True: <span class="math">1 \text{ radian} \approx 57.3°</span>, so <span class="math">\sin 1 = \sin 57.3° \approx 0.841</span>, while <span class="math">\sin 1° \approx 0.0175</span>. Thus <span class="math">\sin 1 > \sin 1°</span>.</p><p>(b) True: <span class="math">\tan 1 = \tan 57.3° \approx 1.557</span>, while <span class="math">\tan 1° \approx 0.0175</span>. Thus <span class="math">\tan 1 > \tan 1°</span>.</p><p>(c) False: <span class="math">4 \text{ radians} \approx 229.2°</span>, which is in the third quadrant where sine is negative. <span class="math">\sin 4 \approx -0.757</span>, while <span class="math">\sin 4° \approx 0.0698</span>. Thus <span class="math">\sin 4 < \sin 4°</span>.</p><p>(d) True: <span class="math">\tan 4 \approx 1.160</span>, while <span class="math">\tan 4° \approx 0.0699</span>. Thus <span class="math">\tan 4 > \tan 4°</span>.</p>
Correct Answer: A, B, D