Download Using Fundamental Identities Introduction Introduction Introduction

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Introduction
We will learn how to use the fundamental
Using Fundamental Identities
Precalculus 5.1
identities to do the following.
1. Evaluate trigonometric functions.
2. Simplify trigonometric expressions.
3. Develop additional trigonometric identities.
4. Solve trigonometric equations.
Introduction
Introduction
Example 1
Example 2
Use the values sin x  12 and cos x  0 to fine
the values of all six trigonometric functions.
cont’d
Simplify cos 2 x csc x  csc x
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Example 3
Factor each expression.
a) 1 cos 2 x
b) 2 csc 2 x  7 csc x  6
Example 5
Simplify csc t  cos t cot t
Example 4
Factor sec 2 x  3 tan x  1
Example 6
Perform the addition and simplify.
1
1

1  sin  1  sin 
Example 7
Rewrite the statement so that it is NOT in
fractional form.
cos 2 y
1  sin y
Example 8
Use the substitution x  5 sin  ,0    2
to write 25  x 2 as a trigonometric function
of  .
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Example 9
Rewrite ln sec   ln cot  as a single logarithm
and simplify the result.
Verifying Trigonometric Identities
Precalculus 5.2
Introduction
In this section, you will study techniques for verifying
trigonometric identities.
Remember that a conditional equation is an equation
that is true for only some of the values in its domain.
For example, the conditional equation
sinx = 0
Introduction
On the other hand, an equation that is true for
all real values in the domain of the variable is
an identity. For example,
the familiar equation
Identity
sin2x = 1 – cos2x
Conditional equation
is true for all real numbers x. So, it is an identity.
is true only for x = n, where n is an integer. When you
find these values, you are solving the equation.
Verifying Trigonometric Identities
Although there are similarities, verifying that a trigonometric
equation is an identity is quite different from solving an
equation. There is no well-defined set of rules to follow in
verifying trigonometric identities, and the process is best
learned by practice.
Example 1
Verify the identity
sin 2   cos 2 
1
cos 2  sec 2 
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Example 2
Verify the identity
1
1

 2 csc 2 
1  cos  1  cos 
Example 4
Verify the identity
csc x  sin x  cos x cot x
Example 6
Verify the identity
2
tan 
1  cos 

1  sec 
cos 
Example 3
Verify the identity
sec
2


x  1 sin 2 x  1   sin 2 x
Example 5
Verify the identity
csc   cot  
sin 
1  cos 
Example 7
Verify each identity
a) tan 3 x  tan x sec 2 x  tan x
b) cos 3 x sin 4 x  (sin 4 x  sin 6 x ) cos x
c) cot 4 x csc x  cot 2 x (csc 3 x  csc x )
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Introduction
To solve a trigonometric equation, use standard algebraic
techniques such as collecting like terms and factoring.
Solving Trigonometric Equations
Your preliminary goal in solving a trigonometric equation
is to isolate the trigonometric function in the equation.
Precalculus 5.3
For example, to solve the equation 2 sin x = 1, divide each
side by 2 to obtain
Introduction
Introduction
To solve for x, note in Figure 5.6 that the equation
has solutions x =  /6 and x = 5 /6 in the interval [0, 2).
Moreover, because sin x has a period of 2, there are
infinitely many other solutions, which can be written as
and
General solution
where n is an integer, as shown in Figure 5.6.
Figure 5.6
Introduction
Another way to show that the equation
many solutions is indicated in Figure 5.7.
Example 1
has infinitely
Solve sin x  2   sin x
Any angles that are coterminal with  /6 or 5 /6 will also be
solutions of the equation.
When solving trigonometric
equations, you should write
your answer(s) using exact
values rather than decimal
approximations.
Figure 5.7
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Example 2
Solve 4 sin 2 x  3  0
Example 4
Find all solutions of the equation on the
interval 0,2 .
Example 3
Solve sin 2 x  2 sin x
Example 5
Solve 3 sec 2 x  2 tan 2 x  4  0
2 sin 2 x  3 sin x  1  0
Example 6
Find all of the solutions of the equation on the
interval 0,2 .
Example 7
Find all the solutions of sin 2 x 
3
2
0
sin x  1  cos x
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Example 8
Find all the solutions of tan 2x  1  0
Example 9
Find all the solutions of 4 tan 2 x  5 tan x  6
Using Sum and Difference Formulas
Sum & Difference Formulas
Precalculus 5.4
Example 1
Find the exact value of cos 12
Example 2
Find the exact value of sin 75
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Example 3
Find the exact value of cos(u  v ) given sin u 
where 0  u  2 , and cos v   45 , where

2
Example 4
7
25
,
Write sin(arctan 1  arccos x ) as an algebraic
expression.
 v  .
Example 5
Prove the cofunction identity sin x  2    cos x
Example 6
Simplify each expression
a) sin  32   
b) tan   4 
Example 7
Find all of the solutions of the equation on the
interval 0,2 .
sin x  2   sin x  32   1
Example 8
Verify that
cos x  h   cos x
 cosh  1 
 sinh 
 cos x
  sin x

h
h


 h 
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Multiple-Angle Formulas
Multiple-Angle &
Product-to-Sum Formulas
You should learn the double-angle formulas because they are
used often in trigonometry and calculus.
Precalculus 5.5
Example 1
Solve cos 2 x  cos x  0
Example 3
Use sin u  35 ,0  u  2 , to find sin 2u , cos 2u ,
and tan 2u.
Example 2
Use a double angle formula to rewrite the
equation g ( x )  3  6 sin 2 x.
Then sketch the graph of the equation over the
interval 0,2 .
Example 4
Derive a triple-angle formula for cos 3 x.
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Power-Reducing Formulas
The double-angle formulas can be used to obtain the following
power-reducing formulas. Example 5 shows a typical power
reduction that is used in calculus.
Half-Angle Formulas
You can derive some useful alternative forms of the
power-reducing formulas by replacing u with u / 2. The results
are called half-angle formulas.
Example 7
Find all solutions of cos 2 x  sin 2
interval 0,2 .
Example 5
Rewrite tan 4 x as a quotient of first powers of
the cosines of multiple angles.
Example 6
Find the exact value of cos 105
Product-to-Sum Formulas
x
2
on the
Each of the following product-to-sum formulas can be verified using
the sum and difference formulas.
Product-to-sum formulas are used in calculus to evaluate integrals
involving the products of sines and cosines of two different angles.
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Example 8
Rewrite sin 5 cos 3 as a sum or difference.
Example 9
Find the exact value of sin 195  sin 105
Example 11
Verify the identity
sin 6 x  sin 4 x
 tan 5 x
cos 6 x  cos 4 x
Product-to-Sum Formulas
Occasionally, it is useful to reverse the procedure and write a
sum of trigonometric functions as a product. This can be
accomplished with the following sum-to-product formulas.
Example 10
Solve sin 4 x  sin 2 x  0
Example 12
Ignoring air resistance, the range of a projectile
fired at an angle  with the horizontal and
with an initial velocity of v0 feet per second is
given by r  161 v0 2 sin  cos  , where r is the
horizontal distance (in feet) that the projectile
will travel. A place kicker for a football team
can kick a football from ground level with an
initial velocity of 78 feet per second.
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Example 12 cont’d
a) At what angle must the player kick the
football so that it travels 188 feet?
b) For what angle is the horizontal distance the
football travels a maximum?
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