Trigonometry involves measuring angles and relationships between sides and angles of triangles. There are six trigonometric ratios - sine, cosine, tangent, cotangent, secant and cosecant - that relate the measures of sides and angles. Angles can be measured in degrees or radians and converted between the two units. Important trigonometric identities relate the ratios to each other and allow trigonometric functions of combined angles to be simplified.

1. Trigonometry

5. • 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.

6. • 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.
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
• 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 .

7. • 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

8. BASIC TRIGONOMETRIC RATIOS
Example:
In a right triangle,
Example:
In a right triangle,
Example:
In the figure below, ΔABC is a right triangle.
Given that AC = 4, BC = 5 and
m(∠ACB) = x, find
Example: In the figure, ΔABC is a right triangle.
Given that AB = 3, AC = 4 and m(∠ACB) = x, find
the six trigonometric ratios for x.

9. TRIGONOMETRIC IDENTITIES
Example:
The trigonometric ratios are related to each
other by equations called trigonometric
identities.
sin x ⋅ cot x ⋅ sec x
Simplify the followings.
1+ sin x
1+ csc x
sin x ·cos x sin x
+
tan x
csc x
1+ sin x
cos x
cos x
1- sin x
(sin x + cos x )2 -1
(sin x - cos x )2 -1
cos x
1- sin x
+
1- sin x
cos x
a2 + c2 = b2
Pythagorean
Trigonometric Ratios of Some Special Angles
theorem
sin x
tan x =
cos x
cot x =
1
csc x =
sin x
1
sec x =
cos x
cos x
sin x
sin2x + cos2x = 1
tan2x + 1 = sec2x
cot2x + 1 = csc2x
tan x ⋅ cot x = 1
0˚
sin
cos
tan
cot
30˚
45˚
60˚
90˚

10. Basic Trigonometric Theorems
Law of Cosine
a2 = b2 + c2 – 2bc ⋅ cos A
b2 = a2 + c2 – 2ac ⋅ cos B
c2 = a2 + b2 – 2ab ⋅ cos C
Law of Sine

11. Sum and Difference Formulas
sin(x + y) = sin x ⋅ cos y + cos x ⋅ sin y
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
tan x + tan y
tan( x + y ) =
1- tan x ·tan y
tan x - tan y
tan( x - y ) =
1+ tan x ·tan y
Example:
cos 75˚ = ?
Example:
sin 105˚ = ?
Example:
tan 75˚ = ?
Example:
sin 15˚ = ?
Example:
cos 120˚ = ?
Example:
sin 135˚ = ?