Understanding Trigonometry

Trigonometry is the study of the relationships between the angles and sides of triangles. It is a fundamental concept in geometry.
Sine (sin), cosine (cos), and tangent (tan) are the basic functions in trigonometry. They are ratios defined in terms of the sides of a right-angled triangle, with the hypotenuse being the longest side.
These ratios can be used to solve a variety of problems in trigonometry, including finding missing angles and lengths in a right-angled triangle.

Trigonometric Ratios

The sine of an angle (sin θ) in a right-angled triangle is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse.
The cosine of an angle (cos θ) is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
The tangent of an angle (tan θ) is the ratio of the sine of the angle to the cosine of the angle, which is equivalent to the ratio of the side opposite to the angle to the side adjacent to the angle.

Sin² θ + cos² θ = 1. This fundamental identity is derived from the Pythagorean theorem and is used frequently in trigonometry.
To compute the sine, cosine or tangent of particular angles (30°, 45°, 60°, and 90°), students should memorise the following ratios as they frequently occur in GCSE standard problems: sin 30°=1/2, cos 30°=√3/2, tan 30°=1/√3; sin 45°=1/√2=√2/2, cos 45°=1/√2=√2/2, tan 45°=1; sin 60°=√3/2, cos 60°=1/2, tan 60°=√3; sin 90°=1, cos 90°=0, tan 90°= undefined.

To solve a trigonometric equation, you may need to use algebraic techniques such as factoring, applying the quadratic formula, or isolating variables.
It is important to check your solutions in the context of the problem. Sometimes extraneous solutions (solutions that do not actually satisfy the original equation) are introduced during the solving process.

Trigonometry has practical applications in various fields including physics, engineering, navigation, music and even computer graphics. It is used extensively in solving real-world problems involving directions, heights and distances.
Problems involving angles of elevation or depression can normally be solved using right-angle trigonometry. The angle of elevation is always measured upwards from the horizontal, and likewise, the angle of depression is always measured downwards.