Plaintext
Math Trigonometry Review
CHSN Review Project
This review guide was written by Dara Adib. It was designed for a test on trigonometry administered
by Jeanine Lennon to her Math 12H (4H/Precalculus) class on February 12, 2008, but also applies to
trigonometry material in Math 11H (3H).
This is a development version of the text that should be considered a work-in-progress.
This review guide and other review material are developed by the CHSN Review Project.
Copyright © 2008-2009 Dara Adib. This is a freely licensed work, as explained in the Definition of Free
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Angles and Sectors of Circles
Radians
π .
• To convert degrees to radians, multiply by 180
• To convert radians to degrees, multiply by 180
π .
• One radian equals 180 ◦
π or approximately 57.296 .
Coterminal Angles
coterminal angles different angles that have the same initial and terminal ray
The differences between coterminal angles are multiples of 360◦ .
Examples: 60◦ , −300◦ , 420◦
Sectors
sector part of a circle formed by two radii and an arc
s = rθ (arc length = radius × angle)
The angle must be represented in radians.
apparent size the angle that an object subtends at one’s eyes; this explains why an object appears to
be smaller when farther away
Trigonometric Functions
Reciprocal Functions
The following are reciprocal functions.
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• csc x =
sin x
1
• sec x =
cos x
1
• cot x =
tan x
As a result, the following are true as well.
1
• sin x =
csc x
1
• cos x =
sec x
1
• tan x =
cot x
2
θ (degrees) θ (radians) Point sin θ cos θ tan θ
0◦ 0 (1, 0) 0 1 0
π
90◦ (0, 1) 1 0 undefined
2
180◦ π (−1, 0) 0 −1 0
3π
270◦ (0, −1) −1 0 undefined
2
θ (degrees) θ (radians) Point cot θ sec θ csc θ
0◦ 0 (1, 0) undefined 1 undefined
π
90◦ (0, 1) 0 undefined 1
2
180◦ π (−1, 0) undefined −1 undefined
3π
270◦ (0, −1) 0 undefined −1
2
Table 1: quadrantal angles
Cofunctions
The following are cofunctions.
• sin x = cos(90 − x)
• tan x = cot(90 − x)
• sec x = csc(90 − x)
As a result, the following are true as well.
• cos x = sin(90 − x)
• cot x = tan(90 − x)
• csc x = sec(90 − x)
Special Angles
Quadrantal Angles
quadrantal angles angles that have a terminal side coinciding with a coordinate axis
The value of the trigonometric function (i.e. sine, cosine, tangent) is determined by the coordinates of
the points on a unit circle. By definition, the point (x, y) on a unit circle corresponds to (cos θ, sin θ).
As a result, quadrantal angles can be determined easily. Table 1 lists quadrantal angles and the values
sin θ
of trigonometric functions. Since tan θ = cos θ , the tangent function of an angle can be found by
dividing the sine function of the angle by the cosine function of the angle.
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Quadrant Reference Angle (β)
I β=θ
II β = 180◦ − θ
III β = θ − 180◦
IV β = 360◦ − θ
Table 2: reference angles
Angles greater than 90◦
Angles greater than 90◦ may be in different quadrants. The trigonometric functions vary over whether
the trigonometric function of a certain angle is positive or negative. It is useful to remember the
mnemonic device “All Students Take Calculus.”
• All trigonometric functions are positive in the first quadrant
• Sine and cosecant are positive in the second quadrant
• Tangent and cotangent are positive in the third quadrant
• Cosine and secant are positive in the fourth quadrant
• The remaining trigonometric functions in each quadrant are negative.
Reference Angles
The use of reference angles is a way to simplify the calculation of the values of trigonometric functions
in different quadrants. Refer to Table 2, where θ is the angle (in degrees) and β is the reference angle.
Sum and Difference Formulas
Sum Formulas
Sine
sin(A + B) = sin A cos B + cos A sin B
Cosine
cos(A + B) = cos A cos B − sin A sin B
Tangent
The derivation for the tangent of the sum of two angles follows.
tan(A + B)
sin(A + B)
cos(A + B)
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sin A cos B + cos A sin B
cos A cos B − sin A sin B
sin A cos B cos A sin B
+
cos A cos B cos A cos B
cos A cos B sin A sin B
−
cos A cos B cos A cos B
tan A + tan B
1 − tan A tan B
Difference Formulas
Sine
sin(A − B) = sin A cos B − cos A sin B
Cosine
cos(A − B) = cos A cos B + sin A sin B
Tangent
The derivation for the tangent of the difference of two angles follows.
tan(A − B)
sin(A − B)
cos(A − B)
sin A cos B − cos A sin B
cos A cos B + sin A sin B
sin A cos B cos A sin B
−
cos A cos B cos A cos B
cos A cos B sin A sin B
+
cos A cos B cos A cos B
tan A − tan B
1 + tan A tan B
Identities
Pythagorean Identities
1. sin2 θ + cos2 θ = 1
2. 1 + tan2 θ = sec2 θ
3. 1 + cot2 θ = csc2 θ
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Quotient Identities
sin θ
1. tan θ =
cos θ
cos θ
2. cot θ =
sin θ
Graphing Trigonometric Functions
General Form
The general form of a trigonometric function is one of the following (sine or cosine).
y = a sin b(x ± c) ± d
y = a cos b(x ± c) ± d
In other words, it can be understood to mean the following.
y = amplitude × sin [frequency (x − horizontal translation)] + vertical translation
y = amplitude × cos [frequency (x − horizontal translation)] + vertical translation
A negative amplitude means a reflection over the x-axis. The vertical translation may be kept in front
only to prevent the ambiguity that the number may be part of the trigonometric function.
Frequency and Period
The following statements are true regarding the relationship between frequency and period.
2π
frequency =
period
2π
period =
frequency
Sine and Cosine
The sine function follows the form zero, maximum, zero, minimum, before any translation.
The cosine function follows the form maximum, zero, minimum, zero, before any translation.
As a result, sine and cosine are horizontal translations of each other.
Double and Half Angle Formulas
Double Angle Formulas
The following are determined by plugging in an angle twice into the sum formulas.
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Sine
sin 2A = 2 sin A cos A
Cosine
cos 2A = cos2 A − sin2 A
More Cosine Formulas However, more formulas for the double of an angle with cosine can be
determined since sin2 A + cos2 A = 1 (first Pythagorean identity).
cos 2A = 1 − 2 sin2 A
cos 2A = 2 cos2 A − 1
Tangent
2 tan A
tan 2A =
1 − tan2 A
Half Angle Formulas
Sine
r
1 1 − cos A
sin A = ±
2 2
Cosine
r
1 1 + cos A
cos A = ±
2 2
Tangent
r
1 1 − cos A
tan A = ±
2 1 + cos A
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