Using Sum and Difference Formulas - Big Ideas Learning
[Pages:6]9.8
Using Sum and Difference Formulas
Essential Question How can you evaluate trigonometric
functions of the sum or difference of two angles?
Deriving a Difference Formula
Work with a partner. a. Explain why the two triangles shown are congruent.
(cos a, sin a) 1
yd
a b
(cos b, sin b)
x
y (cos(a - b), sin(a - b))
d 1
a - b (1, 0)
x
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.
b. Use the Distance Formula to write an expression for d in the first unit circle. c. Use the Distance Formula to write an expression for d in the second unit circle. d. Write an equation that relates the expressions in parts (b) and (c). Then simplify
this equation to obtain a formula for cos(a - b).
Deriving a Sum Formula Work with a partner. Use the difference formula you derived in Exploration 1 to write a formula for cos(a + b) in terms of sine and cosine of a and b. Hint: Use the fact that
cos(a + b) = cos[a - (-b)].
Deriving Difference and Sum Formulas
Work with a partner. Use the formulas you derived in Explorations 1 and 2 to write formulas for sin(a - b) and sin(a + b) in terms of sine and cosine of a and b. Hint: Use the cofunction identities
( ) ( ) sin --2 - a = cos a and cos --2 - a = sin a
and the fact that
[( ) ] cos --2 - a + b = sin(a - b) and sin(a + b) = sin[a - (-b)].
Communicate Your Answer
4. How can you evaluate trigonometric functions of the sum or difference of two angles?
5. a. Find the exact values of sin 75? and cos 75? using sum formulas. Explain your reasoning.
b. Find the exact values of sin 75? and cos 75? using difference formulas. Compare your answers to those in part (a).
Section 9.8 Using Sum and Difference Formulas 519
9.8 Lesson
Core Vocabulary
Previous ratio
What You Will Learn
Use sum and difference formulas to evaluate and simplify trigonometric expressions.
Use sum and difference formulas to solve trigonometric equations and rewrite real-life formulas.
Using Sum and Difference Formulas
In this lesson, you will study formulas that allow you to evaluate trigonometric functions of the sum or difference of two angles.
Core Concept
Sum and Difference Formulas Sum Formulas
sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b - sin a sin b
Difference Formulas sin(a - b) = sin a cos b - cos a sin b cos(a - b) = cos a cos b + sin a sin b
tan(a + b) = -- 1ta-n taan+-- atatannbb
tan(a - b) = -- 1ta+n taan--- atatannbb
In general, sin(a + b) sin a + sin b. Similar statements can be made for the other trigonometric functions of sums and differences.
Check sin(15?) .2588190451 ( (6)- (2))/4 .2588190451
Check tan(7/12) -3.732050808 -2- (3) -3.732050808
Evaluating Trigonometric Expressions Find the exact value of (a) sin 15? and (b) tan -- 712 .
SOLUTION
a. sin 15? = sin(60? - 45?)
Substitute 60? - 45? for 15?.
= sin 60? cos 45? - cos 60? sin 45?
Difference formula for sine
( ) ( ) -- --
--
= -- 23 -- 22 - --21 -- 22
Evaluate.
--
--
= -- 6 -4 2
Simplify.
--
--
The exact value of sin 15? is -- 6 -4 2 . Check this with a calculator.
( ) b. tan -- 712 = tan --3 + --4
= -- 1ta-n t--3an+-- --3tatann--4--4
Substitute --3 + --4 for -- 712 . Sum formula for tangent
--
=
3 + 1 -- 1 - --3 1
= -2 - --3
Evaluate. Simplify.
The exact value of tan -- 712 is -2 - --3. Check this with a calculator.
520 Chapter 9 Trigonometric Ratios and Functions
ANOTHER WAY
You can also use a Pythagorean identity and quadrant signs to find sin a and cos b.
Using a Difference Formula
Find cos(a - 0 < b < --2 .
b)
given
that
cos
a
=
---54
with
<
a
<
-- 32
and
sin
b
=
-- 153
with
SOLUTION
Step 1
Find sin a and cos b.
Because cos a = Quadrant III, sin
---45 a =
and a is ---35, as
in
shown in the figure.
Because sin b = -- 153 Quadrant I, cos b =
and -- 1123,
b is in as shown
in the figure.
y
y
4
a
x
52 - 42 = 3
5
13
b
5
x
132 - 52 = 12
Step 2 Use the difference formula for cosine to find cos(a - b).
cos(a - b) = cos a cos b + sin a sin b Difference formula for cosine
( ) ( )( ) = ---54 -- 1132 + ---53 -- 153
Evaluate.
= --- 6653 The value of cos(a - b) is --- 6635.
Simplify.
Simplifying an Expression
Simplify the expression cos(x + ).
SOLUTION cos(x + ) = cos x cos - sin x sin = (cos x)(-1) - (sin x)(0) = -cos x
Sum formula for cosine Evaluate. Simplify.
Monitoring Progress
Help in English and Spanish at
Find the exact value of the expression.
1. sin 105?
2. cos 15?
3. tan -- 512
4. cos -- 12
5.
Find with
sin(a - b) given < b < -- 32.
that
sin
a
=
-- 187
with
0
<
a
<
--2
and
cos
b
=
--- 2254
Simplify the expression.
6. sin(x + )
7. cos(x - 2)
8. tan(x - )
Section 9.8 Using Sum and Difference Formulas 521
Solving Equations and Rewriting Formulas
ANOTHER WAY
You can also solve the equation by using a graphing calculator. First, graph each side of the original equation. Then use the intersect feature to find the x-value(s) where the expressions are equal.
air
light
prism
Solving a Trigonometric Equation
( ) ( ) Solve sin x + --3 + sin x - --3 = 1 for 0 x < 2.
SOLUTION
( ) ( ) sin x + --3 + sin x - --3 = 1
sin x cos --3 + cos x sin --3 + sin x cos --3 - cos x sin --3 = 1 --12sin x + -- 2--3 cos x + --21sin x - -- 2--3 cos x = 1 sin x = 1
In the interval 0 x < 2, the solution is x = --2 .
Write equation. Use formulas. Evaluate. Simplify.
Rewriting a Real-Life Formula
The index of refraction of a transparent material is the ratio of the speed of light in a vacuum to the speed of light in the material. A triangular prism, like the one shown, can be used to measure the index of refraction using the formula
( ) For
n
= =
-- sin s--2in+--2 --2 . 60?, show that
the
formula
can
be
rewritten
as
n
=
-- 2--3
+
-- 2 1
cot
--2.
SOLUTION
( ) n
=
sin --2 + 30? -- sin-- --2
= -- sin --2 cos 3-- 0?si+n --2cos-- --2 sin 30?
( )( ) ( )( ) --
= -- sin --2 -- -- 23sin+--2 co-- s --2 --21
=
-- 2--3 sin --2 -- sin --2
+
-- --21scinos--2--2
=
--
-- 23
+
-- 1 2
cot
-- 2
Write
formula
with
-- 2
=
-- 620?
=
30?.
Sum formula for sine
Evaluate.
Write as separate fractions. Simplify.
Monitoring Progress
Help in English and Spanish at
( ) ( ) 9. Solve sin --4 - x - sin x + --4 = 1 for 0 x < 2.
522 Chapter 9 Trigonometric Ratios and Functions
9.8 Exercises
Dynamic Solutions available at
Vocabulary and Core Concept Check
1. COMPLETE THE SENTENCE Write the expression cos 130? cos 40? - sin 130? sin 40? as the cosine of an angle.
2. WRITING Explain how to evaluate tan 75? using either the sum or difference formula for tangent.
Monitoring Progress and Modeling with Mathematics
In Exercises 3?10, find the exact value of the expression. (See Example 1.)
3. tan(-15?)
4. tan 195?
5. sin -- 2132 7. cos 105? 9. tan -- 1172
6. sin(-165?)
8. cos -- 1112
( ) 10. sin --- 712
In Exercises 11?16, evaluate the expression given
that cos
--32 < b
a <
= --54
2.
with (See
0 < a < --2 and
Example 2.)
sin
b
=
---1175
with
11. sin(a + b)
12. sin(a - b)
13. cos(a - b)
14. cos(a + b)
15. tan(a + b)
16. tan(a - b)
In Exercises 17?22, simplify the expression. (See Example 3.)
17. tan(x + )
( ) 18. cos x - --2
19. cos(x + 2)
20. tan(x - 2)
( ) 21. sin x - -- 32
( ) 22. tan x + --2
ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in simplifying the expression.
23.
( ) tan x + --4 = -- 1t+antxa+n-- xttaann--4--4
= -- t1a+n xt+an1x
= 1
24.
( ) sin x - --4 = sin --4 cos x - cos --4 sin x
= --2-- 2 cos x - --2-- 2 sin x
= --2-- 2 (cos x - sin x)
25. What are the solutions of the equation 2 sin x - 1 = 0
for 0 x < 2?
A --3
B --6
C -- 23
D -- 56
26. What are the solutions of the equation tan x + 1 = 0 for 0 x < 2?
A --4
B -- 34
C -- 54
D -- 74
In Exercises 27? 32, solve the equation for 0 x < 2. (See Example 4.)
( ) 27. sin x + --2 = --12
( ) 28. tan x - --4 = 0
( ) ( ) 29. cos x + --6 - cos x - --6 = 1
( ) ( ) 30. sin x + --4 + sin x - --4 = 0
31. tan(x + ) - tan( - x) = 0
32. sin(x + ) + cos(x + ) = 0
33. USING EQUATIONS Derive the cofunction identity
( ) sin --2 - = cos using the difference formula
for sine.
Section 9.8 Using Sum and Difference Formulas 523
34. MAKING AN ARGUMENT Your friend claims it is possible to use the difference formula for tangent to
( ) derive the cofunction identity tan --2 - = cot . Is
your friend correct? Explain your reasoning.
35. MODELING WITH MATHEMATICS A photographer is at a height h taking aerial photographs with a 35-millimeter camera. The ratio of the image length WQ to the length NA of the actual object is given by the formula camera
Q
-- WNAQ = -- 35 tan( h-tat-- n) + 35 tan t
h W
t
N
A
where is the angle between the vertical line
perpendicular to the ground and the line from the
camera to point A and t is the tilt angle of the film.
When t = 45?, show that the formula can be rewritten
as -- WNAQ = -- h(1 +7-- 0tan ). (See Example 5.)
36. MODELING WITH MATHEMATICS When a wave travels through a taut string, the displacement y of each point on the string depends on the time t and the point's position x. The equation of a standing wave can be obtained by adding the displacements of two waves traveling in opposite directions. Suppose a standing wave can be modeled by the formula
( ) ( ) y = A cos -- 23t - -- 25x + A cos -- 23t + -- 25x .
When t = 1, show that the formula can be rewritten as y = -A cos -- 25 x .
37. MODELING WITH MATHEMATICS The busy signal on a touch-tone phone is a combination of two tones with frequencies of 480 hertz and 620 hertz. The individual tones can be modeled by the equations:
480 hertz: y1 = cos 960t
620 hertz: y2 = cos 1240t
The sound of the busy signal can be modeled by y1 + y2. Show that y1 + y2 = 2 cos 1100t cos 140t.
( (
38. HOW DO YOU SEE IT? Explain how to use the figure
( ) ( ) to solve the equation sin x + --4 - sin --4 - x = 0
for 0 x < 2.
( y
f(x) = sin
x
+
4
x
2
-1
( g(x) = sin
4
-
x
39. MATHEMATICAL CONNECTIONS The figure shows the
acute angle of intersection, 2 - 1, of two lines with slopes m1 and m2.
y
y = m1x + b1
y = m2x + b2
2 - 1
1
2
x
a. Use the difference formula for tangent to write an equation for tan (2 - 1) in terms of m1 and m2.
b. Use the equation from part (a) to find the acute
angle of intersection of the lines y = x - 1 and
() --
y =
1 -- --3 - 2
x + -- 24 -- --33 .
40. THOUGHT PROVOKING Rewrite each function. Justify your answers. a. Write sin 3x as a function of sin x. b. Write cos 3x as a function of cos x. c. Write tan 3x as a function of tan x.
Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons
Solve the equation. Check your solution(s). (Section 7.5)
41. 1 - -- x -9 2 = ---27
42. -- 1x2 + --43 = --8x
43. -- 2xx+-13 = -- x21-0 1 + 5
524 Chapter 9 Trigonometric Ratios and Functions
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