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Math 150, Fall 2008, Benjamin
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Chapter 7, Continued
7.3 Double-Angle, Half-Angle, and Product-Sum Formulas
Double-Angle Formulas
Formula for Sine:
sin 2x = 2 sin x cos x
Formulas for Cosine:
cos 2x = cos2 x − sin2 x
= 1 − 2 sin2 x
= 2 cos2 x − 1
Formula for Tangent:
tan 2x =
2 tan x
1 − tan2 x
All of these formulas follow directly from the addition formulas for trig functions from the last section. So,
if you remember the addition formulas, you can deduce these if you forget them.
Show why the the formulas for cosine are true using the addition formula for cosine and the Pythagorean
identities.
Example: If sin x = − 35 and x is in Quadrant III, find sin 2x, cos 2x, and tan 2x.
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Math 150, Fall 2008, Benjamin
Aurispa
Formulas for Lowering Powers These definitely will reappear in calculus.
sin2 x =
1 − cos 2x
2
tan2 x =
cos2 x =
1 + cos 2x
2
1 − cos 2x
1 + cos 2x
These formulas follow from the double angle formulas for cosine.
Example: Express sin4 x in terms of the first power of cosine.
Half-Angle Formulas
r
r
u
1 − cos u
1 + cos u
cos = ±
2
2
2
u
1 − cos u
sin u
tan =
=
2
sin u
1 + cos u
u
sin = ±
2
These formulas follow directly from the formulas for lowering powers if you substitute u2 for x and then
take square roots of both sides. (You have to do some simplification to arrive at the half-angle formula for
tangent.)
The choice of sign depends on which quadrant
If 0◦ < u < 90◦ ,
u
2
If 90◦ < u < 180◦ ,
u
2
lies in.
is in which Quadrant?
u
2
is in which Quadrant?
If 180◦ < u < 270◦ ,
u
2
is in which Quadrant?
If 270◦ < u < 360◦ ,
u
2
is in which Quadrant?
Example: Find the exact value of cos 15◦ using a half-angle formula.
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Math 150, Fall 2008, Benjamin
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Example: Given that cot x = −5 and that 270◦ < x < 360◦ , find sin x2 , cos x2 , and tan x2 .
Product-to-Sum Formulas
1
sin u cos v = [sin(u + v) + sin(u − v)]
2
1
cos u sin v = [sin(u + v) − sin(u − v)]
2
1
cos u cos v = [cos(u + v) + cos(u − v)]
2
1
sin u sin v = [cos(u − v) − cos(u + v)]
2
These formulas also follow from the addition formulas.
Verify the third Product-to-Sum formula by using addition formulas.
Example: Find the value of cos 37.5◦ sin 7.5◦ .
Sum-to-Product Formulas
x+y
x−y
cos
2
2
x+y
x−y
sin x − sin y = 2 cos
sin
2
2
x+y
x−y
cos x + cos y = 2 cos
cos
2
2
x+y
x−y
cos x − cos y = −2 sin
sin
2
2
sin x + sin y = 2 sin
These formulas are found by subsituting u =
x+y
2
and v =
3
x−y
2
into the Product-to-Sum formulas.
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Math 150, Fall 2008, Benjamin
Aurispa
Verify the second Sum-to-Product formula by using a Product-to-Sum formula.
π
Example: Find the value of cos 12
+ cos 5π
12 by using a Sum-to-Product Formula.
Verify the identity
sin 3x + sin 7x
= cot 2x.
cos 3x − cos 7x
7.4 Inverse Trigonometric Functions
Remember that only one-to-one functions have inverses. So we cannot find the inverse function of sine,
cosine, or tangent unless we restrict the domain to an interval where the function is one-to-one.
To find the inverse sine function, we restrict the domain of sine to [−π/2, π/2] so that we have a one-to-one
function.
We define the inverse sine function, sin−1 x by sin−1 x = y ↔ sin y = x.
sin−1 x is the number in the interval [−π/2, π/2] whose sine is x.
sin−1 x has domain
and range
.
The inverse sine function is also called arcsine or arcsin.
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Math 150, Fall 2008, Benjamin
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Remember that for inverse functions, we have “cancellation” laws. So, for the sine function
sin(sin−1 x) = x for −1 ≤ x ≤ 1
sin−1 (sin x) = x for − π2 ≤ x ≤
π
2
Examples:
sin−1
√
3
2
sin−1 (− 12 )
sin−1 2
sin−1 (sin π5 )
sin−1 (sin 2π
3 )
The inverse cosine function is very similar to the inverse sine function. In order to have an inverse for cosine,
we restrict the domain of cosine to the interval [0, π].
The inverse cosine function cos−1 is defined by cos−1 x = y ↔ cos y = x.
cos−1 x is the number in the interval [0, π] whose cosine is x.
cos−1 x has domain
and range
.
The inverse cosine function is also called arccosine or arccos.
The same “cancellation” laws hold for cosine.
Examples
cos−1 0
cos−1 (−
√
2
2 )
cos(cos−1 17 )
5
cos−1 (cos 7π
6 )
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Math 150, Fall 2008, Benjamin
Aurispa
The inverse tangent function is also similar. In order to have an inverse for tangent, we restrict the domain
of tangent to the interval (−π/2, π/2).
The inverse tangent function tan−1 is defined by tan−1 x = y ↔ tan y = x.
tan−1 x is the number in the interval (−π/2, π/2) whose tangent is x.
tan−1 x has domain
and range
.
The inverse tangent function is also called arctangent or arctan.
The same “cancellation” laws hold for tangent.
Examples
√
tan−1 (− 3)
tan−1 1
π
tan−1 (tan − 12
)
tan−1 (tan 5π
6 )
In order to compose trig functions with inverse trig functions, the best strategy is to draw a right triangle
with the properties you are given.
Examples: Evaluate the following expressions.
• csc(cos−1
7
25 )
• sin(2 tan−1 35 )
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Math 150, Fall 2008, Benjamin
Aurispa
• cos(2 cot−1 23 )
Write the following as an algebraic expression in x.
• sin(tan−1 x)
• sec(sin−1 x)
7.5 Trigonometric Equations
We can solve trigonometric equations just like any other equation.
• Solve 2 sin x −
√
3 = 0 in the interval [0, 2π).
Since the trig functions are periodic, there are going to be infinitely many solutions, unless we specify
a specific interval that we want x to be in. Recall that sine and cosine have period 2π and tangent
has period π. So what do you do?
– For sine and cosine, solve in [0, 2π), then add 2kπ.
– For tangent, solve in [0, π), then add kπ.
• Find ALL solutions to the above equation.
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Math 150, Fall 2008, Benjamin
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Find all solutions to the following equations.
• sec2 x − 2 = 0
√
• (tan x + 1)( 3 tan x − 1) = 0
• 2 cos2 x + sin x = 1
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Math 150, Fall 2008, Benjamin
Aurispa
If there is a coefficient in front of x, first solve like above, and then divide by the coefficient LAST.
• (a) Find all solutions to the equation 2 cos 3x + 1 = 0. (b) Find the solutions just on the interval
[0, 2π).
• tan5
x
4
− 9 tan x4 = 0
√
• Find all solutions of the equation cos 3x cos x + sin 3x sin x =
9
3
2
on the interval [0, 2π).