Trigonometry and Its Applications
What this domain tests
Trigonometry tests the six ratios, standard-angle values, identities, complementary angles, and heights-and-distances problems using angles of elevation and depression.
Core rules
- In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
- Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.
- Standard values include sin 30° = 1/2, tan 45° = 1, sin 60° = √3/2 and cos 60° = 1/2.
- The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward from the horizontal.
Worked example
From a point 30 m away on level ground, the angle of elevation of the top of a tower is 30°. Find the height. Let h be the height. The horizontal distance is the adjacent side and the height is the opposite side, so tan 30° = h/30. h = 30 × (1/√3) = 10√3 ≈ 17.3 m. Rationalising gives the exact height 10√3 m.
Traps
- Putting the adjacent side in place of the opposite side.
- Measuring the angle of depression from the vertical instead of the horizontal.
- Using the wrong standard-angle value.
- Forgetting to rationalise a denominator such as 1/√3.
- Applying an identity without checking the domain where the ratio is defined.