Heights and distances, basic identities.

Trigonometry studies the relationships between the angles and sides of triangles, especially right triangles. In aptitude tests it is used to evaluate expressions, simplify identities, and find heights and distances.

Trigonometric ratios in a right triangle

• sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent.
• cosec θ = 1 / sin θ, sec θ = 1 / cos θ, and cot θ = 1 / tan θ.
• tan θ = sin θ / cos θ.

Standard values

• sin: 0° = 0, 30° = 1/2, 45° = 1/√2, 60° = √3/2, 90° = 1.
• cos: 0° = 1, 30° = √3/2, 45° = 1/√2, 60° = 1/2, 90° = 0.
• tan: 0° = 0, 30° = 1/√3, 45° = 1, 60° = √3, and tan 90° is not defined.

Basic identities

• sin^2 θ + cos^2 θ = 1.
• 1 + tan^2 θ = sec^2 θ.
• 1 + cot^2 θ = cosec^2 θ.
• Complementary angles: sin(90° - θ) = cos θ, cos(90° - θ) = sin θ, and tan(90° - θ) = cot θ.

Heights and distances

• The angle of elevation is the angle above the horizontal line of sight when looking up at an object. The angle of depression is the angle below the horizontal when looking down.
• Draw a right triangle for the situation and mark the known angle and sides.
• Choose the ratio that connects the known and unknown sides, for example tan θ = height / distance.

How to solve questions

1. Identify whether the question asks for a value, an identity, or a height or distance.
2. Recall the standard values or identity that applies.
3. For heights and distances, draw the figure and pick the right ratio.
4. Simplify carefully and check that the answer is reasonable.