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GCSE Trigonometry Revision

This document provides information about trigonometric identities. It covers fundamental identities like reciprocal, quotient, and Pythagorean identities. It also discusses verifying identities, sum and difference identities for cosine, sine, and tangent, double-angle identities, half-angle identities, and solving problems using various trigonometric identities. Worked examples are provided to illustrate key concepts and steps for solving identity-related problems.

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Prof Samuel Kashina
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102 views36 pages

GCSE Trigonometry Revision

This document provides information about trigonometric identities. It covers fundamental identities like reciprocal, quotient, and Pythagorean identities. It also discusses verifying identities, sum and difference identities for cosine, sine, and tangent, double-angle identities, half-angle identities, and solving problems using various trigonometric identities. Worked examples are provided to illustrate key concepts and steps for solving identity-related problems.

Uploaded by

Prof Samuel Kashina
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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TRIGONOMETRIC IDENTITIES

GCSE 10-12
ADDITIONAL MATHEMATICS
QUICK REVISION

SAMUEL KASHINA (JR)


Trigonometric Identities
5.1 Fundamental Identities

Reciprocal Identities
1 1 1
cot   sec   csc  
tan  cos  sin 

Quotient Identities
sin  cos 
tan   cot  
cos  sin 

Pythagorean Identities
sin 2   cos 2   1 tan 2   1  sec 2 
1  cot 2   csc 2 

Negative-Angle Identities
cos    cos  sec    sec 
sin      sin  csc     csc 
tan      tan  cot      cot 
5
If tan    and θ is in quadrant II:
3
Find sec θ

Find sin θ

Find cot(–θ)

Write cos x in terms of tan x


1  cot 2 
Write in terms of sin θ and cos θ,
1  csc 
2

then simplify the expression so that no


quotients appear.
5.2 Verifying Trigonometric Identities

Hints for Verifying Identities


1. Learn the fundamental identities.
Whenever you see either side of a
fundamental identity, the other side
should come to mind. Also, be aware of
equivalent forms of the fundamental
identities.
2. Try to rewrite the more complicated
side of the equation so that it is identical
to the simpler side.
3. It is sometimes helpful to express
all trigonometric functions in the
equation in terms of sine and cosine
and then simplify the result.
4. Usually, any factoring or indicated
algebraic operations should be
performed. (e.g. squares of sum and
difference, sum and difference of
squares and cubes.)
5. As you select substitutions, keep in
mind the side you are not changing,
because it represents your goal.
6. If an expression contains 1 sin x ,
multiplying both numerator and
denominator by 1 sin x would give
1 sin 2 x , which could be replaced with
cos 2 x .

Verifying identities by changing only one side.

Verify cot   1  csc  cos   sin  


x1  cot x  
2 2 1
Verify tan
1  sin 2 x

tan t  cot t
Verify  sec 2 t  csc 2 t
sin t cos t
cos x 1  sin x
Verify 
1  sin x cos x
Verifying identities by working with both sides.

sec   tan  1  2 sin   sin 2 



sec   tan  cos 2 
Tuners in radios select a radio station by
adjusting the frequency. A tuner may contain
an inductor L and a capacitor, C. The energy
stored in the inductor at time t is given by
Lt   k sin 2 2Ft  and the energy in the
capacitor is given by C t   k cos 2 2Ft ,
where F is the frequency of the radio station
and k is a constant. The total energy E in the
circuit is given by E t   Lt   C t . Verify
the energy is constant.
5.3 Sum and Difference Identities for Cosine

Difference Identity for Cosine


Cosine of a Sum or Difference

cos A  B   cos A cos B  sin A sin B


cos A  B   cos A cos B  sin A sin B

Find cos15

5
Find cos
12

Find cos 87 cos 93  sin 87 sin 93


cos90    

Cofunction Identities
sin   cos90    cos   sin 90   

sec   csc90    csc   sec90   

tan   cot 90    cot   tan90   

Find a value of θ such that cot   tan 25

Find a value of θ such that sin   cos 30


3
Find a value of x such that csc  sec x
4

Write cos180    as a function of θ alone

3
Find coss  t , given that sin s  ,
5
12
cos t   , and both s and t are in quadrant II
13
Alternating current implies that current
alternates direction in the wires. The
voltage in a typical 115V outlet cam be
expressed by V t   163 sin t , ω is angular
speed (radians/sec) of the generator at the
plant and t is time in seconds.

Determine ω at 60 cycles per second.

Determine a value of  so that the graph of


V t   163 cost    is the same as the
graph of V t   163 sin t .
 3 
Verify sec  x    csc x
 2 
5.4 Sum and Difference Identities for Sine
and Tangent

Sum and Difference Identities for Sine

sin  A  B   cos90   A  B  

sin A  B   sin A   B  
Sine of a Sum or Difference

sin A  B   sin A cos B  cos A sin B


sin  A  B   sin A cos B  cos A sin B

sin  A  B 
tan  A  B   
cos A  B 
Tangent of a Sum or Difference
tan A  tan B
tan  A  B  
1  tan A tan B
tan A  tan B
tan  A  B  
1  tan A tan B

Find sin 75

7
Find tan
12

Find sin 40 cos160  cos 40 sin 160


Write sin 30    as a function of θ

Write tan 45    as a function of θ

Write sin 180    as a function of θ


Suppose A and B are angles in standard
4 
position, with sin A  ,  A   , and
5 2
5 3
cos B   ,   B  .
13 2

Find sin  A  B 

Find tan  A  B 
the quadrant of A  B

     
Verify sin      cos     cos 
6  3 
5.5 Double-Angle Identities

cos 2 A  cos A  A 
sin 2 A  sin  A  A 

tan 2 A  tan  A  A 
Double-Angle Identities
cos 2 A  cos 2 A  sin 2 A cos 2 A  1  2 sin 2 A
cos 2 A  2 cos 2 A  1 sin 2 A  2 sin A cos A
2 tan A
tan 2 A 
1  tan 2 A

3
Given cos   and sin   0 , find sin 2
5

3
Given cos   and sin   0 , find cos 2
5
3
Given cos   and sin   0 , find tan 2
5

Find the values of the six trig fcns of θ if


4
cos 2  , 90    180
5
Verify cot x sin 2 x  1  cos 2 x

Simplify cos 2 7 x  sin 2 7 x

Simplify sin 15 cos15


Write sin 3x in terms of sin x

V2
The formula for wattage is W  , V is
R
voltage, R is resistance in ohms.
From last section, V  163 sin 120t

Find max and min wattage.


Product-to-Sum Identities

cos A  B   cos A cos B  sin A sin B


cos A  B   cos A cos B  sin A sin B

sin A  B   sin A cos B  cos A sin B


sin  A  B   sin A cos B  cos A sin B
Product-to-Sum Identities

1
cos A cos B  cos A  B   cos A  B 
2
1
sin A sin B  cos A  B   cos A  B 
2
1
sin A cos B  sin  A  B   sin  A  B 
2
1
cos A sin B  sin  A  B   sin  A  B 
2

Write 4 cos 75 sin 25 as a sum or difference


of two functions.
Sum-to-Product Identities

 A B  A B
sin A  sin B  2 sin   cos 
 2   2 
 A B  A B
sin A  sin B  2 cos  sin  
 2   2 
 A B  A B
cos A  cos B  2 cos  cos 
 2   2 
 A B  A B
cos A  cos B  2 sin   sin  
 2   2 

Write sin 2  sin 4 as a product of two


functions
5.6 Half-Angle Identities

cos 2 x  1  2 sin 2 x
Half-Angle Identities

A 1  cos A A 1  cos A
cos   sin  
2 2 2 2

A 1  cos A
tan  
2 1  cos A

A sin A A 1  cos A
tan  tan 
2 1  cos A 2 sin A

Find the value of cos 15°


Find the value of tan 22.5°
A sin A
tan 
2 1  cos A

2 3 s
Given cos s  ,  s  2 , find sin ,
3 2 2
s s
cos , and tan
2 2
1  cos12 x
Simplify 
2

1 cos 5
Simplify
sin 5
2
 sin x  cos x   1  sin x
Verify  
 2 2 

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