Introduction to Trigonometric Functions

• The key to understanding trigonometry in triangles is sin θ, cos θ, and tan θ. These functions represent the ratios of different pairings of the sides in a right triangle.
• The angle θ is one of the acute angles in the right-angled triangle.

Sin θ

• sin θ is the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. It is mathematically expressed as sin θ = Opposite/Hypotenuse.
• SOH is a useful mnemonic to remember this, which stands for Sine = Opposite/Hypotenuse.

Cos θ

• cos θ is the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. It is mathematically defined as cos θ = Adjacent/Hypotenuse.
• A simple way to remember this is the mnemonic CAH, which stands for Cosine = Adjacent/Hypotenuse.

Tan θ

• tan θ is the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. It is defined as tan θ = Opposite/Adjacent.
• The mnemonic TOA can help remember this relationship: Tangent = Opposite/Adjacent.

Relationships between Trigonometric Functions

• The trigonometric functions are interconnected. For example, the tangent of an angle θ in a right-angled triangle is equal to the sine of the angle divided by the cosine of the angle: tan θ = sin θ/cos θ.
• Additionally, the Pythagorean Trig Identity connects sine and cosine: sin² θ + cos² θ = 1.

Applying Definitions of Trigonometric Functions

• These definitions and relationships of sine, cosine, and tangent can be used to solve many geometry problems. Often, they are employed to find missing side lengths or unknown angles in right-angled triangles.
• Trigonometry plays a crucial role in a wide array of mathematical and real-world applications, from calculating distances and heights to understanding periodic phenomena such as sound and light waves.