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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˚ = ?