Trigonometry

Trigonometry studies the relationship between the angles and side lengths of triangles, especially right-angled triangles.

Trigonometric Ratios: In a right-angled triangle, relative to an angle \( \theta \):
- Sine (sin θ): Opposite / Hypotenuse
- Cosine (cos θ): Adjacent / Hypotenuse
- Tangent (tan θ): Opposite / Adjacent
The other three ratios are reciprocals: Cosecant (csc θ) = 1/sin θ, Secant (sec θ) = 1/cos θ, and Cotangent (cot θ) = 1/tan θ.
Heights and Distances: This is a practical application of trigonometry.
- Angle of Elevation: The angle measured upwards from the horizontal to view an object.
- Angle of Depression: The angle measured downwards from the horizontal to view an object.

Pythagorean Identities:
- \( \sin^2\theta + \cos^2\theta = 1 \)
- \( 1 + \tan^2\theta = \sec^2\theta \)
- \( 1 + \cot^2\theta = \csc^2\theta \)
Values for Standard Angles (0°, 30°, 45°, 60°, 90°): You must memorize the sin, cos, and tan values for these angles.

Mnemonic: Use 'SOH CAH TOA' to remember the primary ratios (Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent).
Heights & Distances: Drawing a clear, labeled diagram is the most critical step. Often, the problem reduces to a simple application of tan θ.
Special Triangles: For angles 30°, 60°, and 45°, you can use the side ratios of special right triangles (1:√3:2 for 30-60-90 and 1:1:√2 for 45-45-90) to solve problems without using sin/cos/tan.

Master the three Pythagorean identities and the table of standard angle values. They are essential.
For identity-based multiple-choice questions, you can sometimes substitute a standard angle like 45° or 60° into the expression and the options to find the correct match.
In Heights and Distances, the Angle of Depression from point A to B is equal to the Angle of Elevation from point B to A.