if tanθ = $frac{3}{4}$, 180° < θ < 270°, find the value of cosθ.

Step 1: Analyze the question We're given tanθ = 3/4 and θ is between 180° and 270° (the third quadrant). We need cosθ.

Step 2: Concept

• tanθ = opposite/adjacent, so we can build a right-angled triangle.
• In the third quadrant, both sine and cosine are negative, and tangent is positive (this matches tanθ = 3/4 > 0).
• Remember "ASTC": All, Sine, Tangent, Cosine. The third quadrant is the "T" quadrant, where only tan is positive.

Step 3: Solution Take opposite = 3 and adjacent = 4.

Hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5

The reference triangle is the 3-4-5 triangle, so:

• cos (reference) = adjacent/hypotenuse = 4/5
• sin (reference) = opposite/hypotenuse = 3/5

Because θ is in the third quadrant, cosine is negative:

cosθ = -4/5

Check: sinθ = -3/5, so tanθ = sinθ/cosθ = (-3/5)/(-4/5) = 3/4 ✓

Step 4: Answer The correct option is C. -4/5.

Tip: When a trig question gives a quadrant range, first find the magnitude using the 3-4-5 (or other Pythagorean) triangle, then apply the sign using ASTC. Here, 180° to 270° means only tan is positive, so sin and cos are both negative.