Tayarri LCAT
LCAT Math, Geometry and Trigonometry

Trigonometry for the LCAT: special angles and quadrants

Without a calculator, LCAT trigonometry relies on two things: the special angles (30°, 45°, 60°) from the special right triangles, and the sign of each function in each quadrant. Sine is positive in Quadrants I and II, cosine in I and IV, and tangent in I and III.

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No calculator on the LCAT. LUMS’s official sample says calculators are not allowed, so every practice question here can be solved by hand.
Anglesincostan
30° (π/6)1/2√3/2√3/3
45° (π/4)√2/2√2/21
60° (π/3)√3/21/2√3
QuadrantPositive functions
I (0°–90°)All
II (90°–180°)sin
III (180°–270°)tan
IV (270°–360°)cos

The LUMS sample question

If tan θ = 3/4 and sin θ < 0, what is cos θ?

tan positive and sin negative → Quadrant III
3-4-5 triangle → cos = 4/5 before the sign
cos is negative in Quadrant III → cos θ = −4/5

Common mistakes

Wrong signFind the quadrant first, then the sign.
Wrong sidetan = opposite/adjacent; build the triangle before choosing.
RadiansReplace π with 180°: 7π/6 = 210°.

Sample questions with answers

1. If tan θ = 3/4 and sin θ < 0, what is cos θ?
  1. 4/5
  2. −4/5 (correct)
  3. −3/5
  4. 3/5

Quadrant III: cosine is negative (LUMS sample).

2. What is sin 150°?
  1. −1/2
  2. 1/2 (correct)
  3. √3/2
  4. −√3/2

Reference angle 30°, sine positive in Quadrant II.

3. In which quadrant is sin θ > 0 and cos θ < 0?
  1. I
  2. II (correct)
  3. III
  4. IV

Only sine is positive in Quadrant II.

Frequently asked questions

Is trigonometry on the LCAT?

Yes. LUMS’s guidelines list trigonometry, and its sample includes a quadrant-sign question.

How do I remember the signs in each quadrant?

All, Sin, Tan, Cos for Quadrants I to IV: "All Students Take Calculus".

Do I need to memorize special-angle values?

Yes, since calculators are not allowed. Derive them from the 30-60-90 and 45-45-90 triangles.

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