Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<p>If <span class="math inline">\(\tan x = -\frac{4}{3}\)</span>, <span class="math inline">\(\frac{3\pi}{2} < x < 2\pi\)</span>, find the value of <span class="math inline">\(9\sec^2 x - 4\cot x\)</span>.</p>
Step-by-Step Solution
Key Concept: Use the identity sec²x = 1 + tan²x to find sec²x from the given tan x value, then determine the correct signs of sec x and cot x using the quadrant information. The angle lies in the third quadrant where both sine and cosine are negative.
<p><strong>Step 1:</strong> Find sec²x using the identity sec²x = 1 + tan²x.</p><p>Given: tan x = -4/3</p><p>sec²x = 1 + tan²x = 1 + (-4/3)² = 1 + 16/9 = 25/9</p><p><strong>Step 2:</strong> Determine the sign of sec x using the given range.</p><p>The condition 3π/2 < x < 2π indicates the fourth quadrant, where cos x > 0, so sec x > 0.</p><p>Therefore: sec x = +5/3</p><p><strong>Step 3:</strong> Find cot x from tan x.</p><p>cot x = 1/tan x = 1/(-4/3) = -3/4</p><p><strong>Step 4:</strong> Calculate 9sec²x - 4cot x.</p><p>9sec²x - 4cot x = 9(25/9) - 4(-3/4)</p><p>= 25 + 3</p><p>= 28</p><p><strong>∴ Answer: 28</strong></p>
Correct Answer: 28