Trigonometry and Its Applications
What the domain tests
This domain tests trigonometric ratios, standard-angle values, identities, and solving height-and-distance problems using angles of elevation and depression.
Core rules and concepts
- In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Reciprocal ratios: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
- Identity: sin²θ + cos²θ = 1; dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.
- Standard values: sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, sin 60° = √3/2.
- Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal.
Worked example
A tower stands on level ground. From a point 30 m from its foot, the angle of elevation of the top is 30°. Let the height be h. Then tan 30° = h/30. Since tan 30° = 1/√3, h = 30/√3 = 10√3 m ≈ 17.32 m.
Traps that cost marks
Do not use the slant distance as the adjacent side. In depression problems, the angle at the observer equals the alternate angle at the object, so redraw the right triangle. Keep exact surd values until the final step. Check that your calculator is in degree mode if you use one.