Trigonometry Fundamentals and Identities Quiz

Trigonometry

Trigonometry is a branch of mathematics that deals with relationships between angles and lengths of sides in a right-angled triangle. The field also includes values derived from those relationships, such as sine, cosine, and tangent. This branch of mathematics plays a crucial role in various aspects of physics, engineering, and computer science.

Sine, Cosine, Tangent

Sine

The sine function assigns to each angle θ in a right triangle the ratio of the length of the side opposite that angle to the longest side of the triangle (the hypotenuse).

Finding Sine

To find the sine of an angle θ in a right triangle, divide the length of the side opposite by the length of the hypotenuse:

sin(θ) = (opposite side)/(hypotenuse)

Cosine

The cosine function assigns to each angle θ in a right triangle the ratio of the length of the adjacent side to the longest side (the hypotenuse).

Finding Cosine

To find the cosine of an angle θ in a right triangle, divide the length of the side adjacent by the length of the hypotenuse:

cos(θ) = (adjacent side)/(hypotenuse)

Tangent

The tangent function assigns to each angle θ in a right triangle the ratio of the length of the opposite side to the length of the adjacent side. It is defined as:

tan(θ) = (opposite side)/(adjacent side)

Finding Tangent

To find the tangent of an angle θ in a right triangle, divide the length of the side opposite by the length of the side adjacent:

tan(θ) = (opposite side)/(adjacent side)

Trigonometric Identities

Trigonometry has many useful formulas called "identities," which are true for all angles θ and radii r:

• sin²θ + cos²θ = 1
• tan²θ + 1 = sec²θ
• cot²θ + 1 = cosec²θ
• sin³θ + cos³θ = 1/2
• sin(θ + θ') = sin(θ)cos(θ') + cos(θ)sin(θ')
• cos(θ + θ') = cos(θ)cos(θ') - sin(θ)sin(θ')
• tan(θ + θ') = (tan(θ) + tan(θ'))/(1 - tan(θ)tan(θ'))

Solving Triangles

Solving triangles involves applying different combinations of the three fundamental functions (sine, cosine, and tangent) to obtain equations that can be solved simultaneously to determine one or more of the unknown side or angle measures. In this process, you first identify the known quantities, then apply appropriate trigonometric ratios to get the desired unknown.

For example, if we have two acute angles and one of the sides (either one of the non-hypotenuse), we can solve the triangle using two trigonometric ratios. We usually solve for one of the angles, say θ:

sin(θ) = (opposite side) / (hypotenuse) cos(θ) = (adjacent side) / (hypotenuse)

Given the information, we can create equations:

sin(θ) = (a) / b cos(θ) = (b) / (c)

By multiplying these two equations together, we get:

sin(θ) * cos(θ) = 1/2

Substituting the expressions for sin(θ) and cos(θ) into the equation and simplifying, we get:

(a^2) / (bc) = 1/2

Which leads to the Law of Cosines:

a² = b² + c² - 2bc*cos(θ)

This allows us to solve for the remaining unknown side length when given the measurements of the other two sides and one angle.