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INVESTIGATE the Math
The cosine of the compound angle can be expressed in terms of the sines and cosines of a and b. Consider the following unit circle diagram:
By the cosine law,
However, c has endpoints of and By the distance formula, Squaring both sides,
Equating and ,
Solving for
How can other formulas be developed to relate the primary trigonometric ratios of a compound angle to the trigonometric ratios of each angle in the compound angle?
?
cos ( a 2 b ) 5 sin a sin b 1 cos a cos b
cos ( a 2 b ),
2 2 2 cos ( a 2 b ) 5 2 2 2 sin a sin b 2 2 cos a cos b
1 2
2 c^2 5 2 2 2 sin a sin b 2 2 cos a cos b
c^2 5 1 2 2 sin a sin b 2 2 cos a cos b 1 1
c^2 5 sin^2 a 1 cos 2 a 2 2 sin a sin b 2 2 cos a cos b 1 sin 2 b 1 cos 2 b
c^2 5 sin^2 a 2 2 sin a sin b 1 sin 2 b 1 cos 2 a 2 2 cos a cos b 1 cos 2 b
c^2 5 (sin^ a^^2 sin^ b )^2 1 (cos^ a^^2 cos^ b )^2
c 5 "(sin a 2 sin b )^2 1 (cos a 2 cos b )^2
(cos a , sin a ) (cos b , sin b ).
1 c^2 5 2 2 2 cos ( a 2 b )
c^2 5 12 1 12 2 2 ( 1 )( 1 )cos ( a 2 b )
y
x 0
a b
c a – b (cos^ b , sin^ b )
–1 1
(cos a , sin a ) 1
( a 2 b )
394 7.2^ Compound Angle Formulas NEL
7.2 Compound Angle Formulas
GOAL Verify and use compound angle formulas.
compound angle an angle that is created by adding or subtracting two or more angles
NEL Chapter 7 395
A. Use a calculator and the special triangles to verify that the subtraction
formula for cosine works if and Repeat for and
B. Use the subtraction formula for cosine to obtain an addition formula for cosine, as follows: i) Rewrite the compound angle equation for. ii) Replace b with and derive an equation for iii) Simplify this equation, using your knowledge of even and odd functions, to write sin in terms of and in terms of
C. Use a calculator and the special triangles to verify your addition
formula for cosine if and
D. To find an addition formula for sine, use the cofunction
identity
i) Write ii) Use the subtraction formula for cosine to expand and simplify this formula.
E. Use a calculator and the special triangles to verify your addition
formula for sine by substituting and
F. Determine and verify a subtraction formula for sine, using the addition formula you found in part D and the strategy you used in part B.
G. Recall that Use this identity to determine addition and
subtraction formulas for and Use a calculator and the special triangles to verify your formulas if and
H. Make a list of all the compound angle formulas that you determined.
Reflecting
I. How did you use equivalent trigonometric expressions to simplify formulas in parts B, D, F, and G?
J. How did you use the special triangles to verify the addition and subtraction formulas you determined?
b 5 p 4.
a 5 p 6
tan ( a 1 b ) tan ( a 2 b ).
tan u 5 (^) cossin^ uu.
sin ( a 2 b ),
a 5 p 3 b 5 p 4.
sin ( a 1 b ) 5 cos Qp 2 2 ( a 1 b )R 5 cos QQp 2 2 a R 2 b R.
sin u 5 cos Q
p 2 2 uR.
sin ( a 1 b ),
b 5 p a 5 4. p 3
cos b.
( 2 b ) sin b , cos ( 2 b )
( 2 b ), cos ( a 1 b ).
cos ( a 2 b )
cos ( a 1 b ),
b 5 p
a 5 p a 5 45 ° b 5 30 °. 3
EXAMPLE 2 Using compound angle formulas to simplify trigonometric expressions
Simplify each expression.
a)
b)
Solution
a)
b)
By expressing an angle as a sum or difference of angles in the special triangles, exact values of other angles can be determined.
5 sin x
5 sin ( 2 x 2 x )
sin 2 x cos x 2 cos 2 x sin x
5 (sin a ) (cos b ) 2 (cos a ) (sin b )
sin ( a 2 b )
5
" 3 2
5 cos p 6
5 cos a 7 p 12
2 5 p 12
b
cos 7 p 12
cos 5 p 12
1 sin 7 p 12
sin 5 p 12
5 (cos a ) (cos b ) 1 (sin a ) (sin b )
cos ( a 2 b )
sin 2 x cos x 2 cos 2 x sin x
cos
7 p 12 cos
5 p 12 1 sin
7 p 12 sin
5 p 12
The expression given is the right side of the subtraction formula for cosine, where and b 5 512 p.
a 5 712 p
The expression given is the right side of the subtraction formula for sine, where a 5 2 x and b 5 x.
Use a special triangle to evaluate cos p 6.
5 p 6
7 p 12 2 5 p 12 5 2 p 12
Chapter 7
Compound angle formulas can be used, both forward and backward, to evaluate and simplify trigonometric expressions.
EXAMPLE 3 Calculating trigonometric ratios of compound angles
Evaluate where a and b are obtuse angles; and
Solution
sin b 5 135.
sin a 5 3 sin ( a 1 b ), 5
and
Sketch each angle in standard position.
5 2
56 65
5 2 36 65 2 20 65
5 a 3 5
b a 2 12 13
b 1 a 2 4 5
b a 5 13
b
sin ( a 1 b ) 5 (sin a ) (cos b ) 1 (cos a ) (sin b )
cos b 5 x r 5 2^
12 13 cos a 5 x r 5 2^
4 5
x 5 2 4 x 5 2 12
x 5 6" 16 x 5 6" 144
x^2 5 25 2 9 x^2 5 169 2
x^2 1 32 5 52 x^2 1 52 5
x^2 1 y^2 5 r^2 x^2 1 y^2 5 r^2
sin b 5 5 13
5 y sin a (^5) r 3 5
5 y r
Use the Pythagorean theorem to determine the x -coordinate of each point on the terminal arm. Since a and b are obtuse angles, their terminal arms lie in the second quadrant, where and In the second quadrant, x must be negative.
p 2 ,^ b^ , p.
p 2 ,^ a^ , p
To determine the sine and cosine of both a and b are required. Determine the cosine of a and b.
sin ( a 1 b ),
Substitute the required trigonometric ratios into the compound angle formula for and then evaluate.
sin ( a 1 b ),
r = 5 y = 3 a
x = –
P (–4, 3)
x
y
r = 13 y = 5 b x = –
x
y P (–12, 5)
Compound angle formulas can also be used to prove the equivalence of trigonometric expressions.
7.2 Compound Angle Formulas
CHECK Your Understanding
1. Rewrite each expression as a single trigonometric ratio. a) b) 2. Rewrite each expression as a single trigonometric ratio, and then evaluate the ratio.
a)
b)
3. Express each angle as a compound angle, using a pair of angles from the special triangles.
a) c) e) 105°
b) d) f )
4. Determine the exact value of each trigonometric ratio.
a) c) e)
b) d) f )
PRACTISING
5. Use the appropriate compound angle formula to determine the exact value of each expression.
a) c) e)
b) d) f )
6. Use the appropriate compound angle formula to create an equivalent expression. a) c) e)
b) d) f )
7. Use transformations to explain why each expression you created in question 6 is equivalent to the given expression.
cos a x 1 tan ( x 1 p) tan ( 2 p 2 x )
3 p 2
b
cos a x 1 sin ( x 2 p)
p 2
sin (p 1 x ) b
cos a
p 2
p 3
sin a 2 b
p 2
p 3
cos ap 2 b
p 4
b
tan a
p 3
p 6
tan a b
p 4
sin ap 1 1 pb
p 6
b
tan
23 p 12
sin a 2
p 12
cos 15 ° b
tan cos 105 °
5 p 12
sin 75 °
5 p 6
p 12
p 6
cos
5 p 12
cos
p 12
1 sin
5 p 12
sin
p 12
tan 170 ° 2 tan 110 ° 1 1 tan 170 ° tan 110 °
cos 4 x cos 3 x 2 sin 4 x sin 3 x
sin a cos 2 a 1 cos a sin 2 a
K
7.2 Compound Angle Formulas
8. Determine the exact value of each trigonometric ratio.
a) c) e)
b) d) f )
9. If and evaluate a) c) e) b) d) f ) 10. and are acute angles in quadrant I, with and Without using a calculator, determine the values of and 11. Use compound angle formulas to verify each of the following cofunction identities.
a) b)
12. Simplify each expression.
a) b)
13. Simplify 14. Create a flow chart to show how you would evaluate
given the values of and if both a and.
15. List the compound angle formulas you used in this lesson, and look for similarities and differences. Explain how you can use these similarities and differences to help you remember the formulas.
Extending
16. Prove 17. Determine in terms of and 18. Prove 19. Prove cos C 2 cos D 5 22 sin a
C 1 D
b sin a
C 2 D
b.
cos C 1 cos D 5 2 cos a
C 1 D
b cos a
C 2 D
b.
cot ( x 1 y ) cot x cot y.
sin C 1 sin D 5 2 sin a
C 1 D
b cos a
C 2 D
b.
sin a sin b , b PS0, p 2 T
cos ( a 1 b ),
sin ( f 1 g ) 1 sin ( f 2 g ) cos ( f 1 g ) 1 cos ( f 2 g ).
cos a x 1
p 3
b 2 sin a x 1
p 6
sin (p 1 x ) 1 sin (p 2 x ) b
cos x 5 sin a
p 2
sin x 5 cos a 2 x b
p 2
2 x b
sin (a 1 b) tan (a 1 b).
cos b 5 135.
a b sin a 5 257
sin ( x 1 y ) sin ( x 2 y ) tan ( x 2 y )
cos ( x 1 y ) cos ( x 2 y ) tan ( x 1 y )
3 p 0 , x , 2 , y , 2 p, p sin y 5 2 2 , 12 sin x 5 13 , 4 5
tan
25 p 12
sin
13 p 12
tan ( 215 °)
tan
7 p 12
cos
11 p 12
cos 75 °
A
T
C
Chapter 7
