Trigonometry

• Before we begin our study of trigonometry, it will be helpful to review these basic concepts and definitions.

• An angle is the union of two rays which have the same initial point.

• If a directed angle is measured in a clockwise direction from its initial side then the angle is a negative angle . If the angle is measured in a counterclockwise direction then it is a positive angle.

• We can measure angles using different units of measurement. The most common units are degree and radian . We write ˚ to show a degree measurement: one full circle measures 360°. We write R to show a radian measurement: one full circle measures 2πR.

• If two or more angles in standard position (its vertex is at the origin of the plane and its initial side lies along the positive x-axis.) have coincident terminal sides then they are called coterminal angles . For example, 90° and -270° are coterminal angles. 180° and -180° are also coterminal angles.

• Let β be an angle which is greater than 360° or less than 0°. Then α is called the a primary directed angle of β if α is coterminal with β and α [0°, 360°). ∈ In other words, α is the angle between 0° and 360° which is coterminal with β.

We can write: β = α ± k · 360° or β = α ± 2kπ .

Example: convert the following degree measurements to radian.a) 180˚ b) 90˚ c) 150˚ d) 120˚ e) 45˚ f) 30˚ g) 60˚

Example: convert the following radian measurements to degree.a) 2π/3 b) π/6 c) 5π/3 d) 7π/4 e) 2π f) π/18 g) 5π/36

• The circle whose center lies at the origin of the coordinate plane and whose radius is 1 unit is called the unit circle.

• The coordinate axes divide the unit circle into four parts, called quadrants. The quadrants are numbered in a counterclockwise direction.

Examples: In which quadrant does each angle lie?a) 75° b) 228° c) 305° d) 740° e) –442° f) 7π/3 g) – 17π/5

BASIC TRIGONOMETRIC RATIOS

Example: In the figure, ΔABC is a right triangle. Given that AB = 3, AC = 4 and m(∠ACB) = x, findthe six trigonometric ratios for x.

Example:

Example:

In a right triangle,

Example:

In a right triangle,

In the figure below, ΔABC is a right triangle. Given that AC = 4, BC = 5 and m(∠ACB) = x, find

TRIGONOMETRIC IDENTITIES

The trigonometric ratios are related to each other by equations called trigonometric identities.

a2 + c2 = b2 Pythagorean theorem

sin2x + cos2x = 1 tan2x + 1 = sec2x

cot2x + 1 = csc2x tan x ⋅ cot x = 1

sin x cot ⋅ x sec ⋅ x

Example: Simplify the followings.

Trigonometric Ratios of Some Special Angles

0˚ 30˚ 45˚60˚ 90˚sin

cos

tan

cot

Basic Trigonometric TheoremsLaw of Cosine

a2 = b2 + c2 – 2bc cos ⋅ Ab2 = a2 + c2 – 2ac cos ⋅ Bc2 = a2 + b2 – 2ab cos ⋅ C

Law of Sine

sin(x + y) = sin x cos ⋅ y + cos x sin ⋅ ySum and Difference Formulas

sin(x – y) = sin x cos ⋅ y – cos x sin ⋅ y

cos(x + y) = cos x cos ⋅ y – sin x sin ⋅ y

cos(x – y) = cos x cos ⋅ y + sin x sin ⋅ y

cos 75˚ = ?Example:

sin 105˚ = ?Example:

tan 75˚ = ?Example:

sin 15˚ = ?Example:

cos 120˚ = ?Example:

sin 135˚ = ?Example: