Download Trigonometric Identities Student Notes

TRIGONOMETRIC IDENTITIES
1. Reciprocal, Quotient, and Pythagorean Identities – pg. 290-297
Assignment: pg. 296-297 #1-8(odd in each), 10, 11, 13
2. Sum, Difference, and Double-Angle Identities – pg. 299-308
Assignment: pg. 306-308 #1(odd), 2(odd), 4(odd), 6, 7, 8(odd), 9-11, 12a, 15a, 16a, 17
3. Proving Identities – pg. 309-315
Assignment: pg. 314-315 #1-3(odd in each), 4-7, 10b, 11b, 12, 13
4. Solving Trigonometric Equations Using Identities – pg. 316 – 321
Assignment: pg. 320-321 #1-11 (odd in each)
5. Chapter Quiz
6. Chapter Review – pg. 322-323
Assignment: pg. 322-323 #1-21(odd in each), omit: 5, 10, 11, 14, 16
7. Chapter Exam
LESSON 1: RECIPROCAL, QUOTIENT, AND PHYTHAGOREAN IDENTITIES
Learning Outcomes:

Learn to verify a trigonometric identity numerically and graphically using technology

Learn to explore reciprocal, quotient, and Pythagorean identities

To determine non-permissible values of trigonometric identities

To explain the difference between a trigonometric identity and a trigonometric equation
What is the difference between an equation and an identity?
2𝑥 2 + 3 = 11 is an equation. It is only true for certain values of the variable x. The solutions to
this equation are -2 and 2 which can be verified by substituting these values into the equation.
(𝑥 + 1)2 = 𝑥 2 + 2𝑥 + 1 is an identity. It is true for all values of the variable x.
Comparing Two Trigonometric Expressions
1. Graph the curves y = sin x and y = cos x tan x over the domain −360° ≤ 𝑥 < 360°. Graph
the curves on separate grids using the same range and scale. What do you notice?
2. Use the table function on your calculator to compare the two graphs. Describe your findings.
3. Use your knowledge of tan x to simplify the expression cos x tan x
2
4. What are the non-permissible values of x in the equation sin x = cos x tan x?
sin 𝑥
Given tan 𝑥 = cos 𝑥
The equations y = sin x and y = cos x tan x are examples of trigonometric identities.
Trigonometric identities can be verified both numerically and graphically.
Reciprocal Identities:
csc x 
1
sin x
sec x 
1
cos x
cot x 
1
tan x
Quotient Identities:
tan x 
sin x
cos x
cot x 
cos x
sin x
Verifying Identities for a Particular Case:
When verifying an identity we must treat the left side (LS) and the right side (RS) separately and
work until both sides represent the same value.
This technique does not prove that an identity is true for all values of the variable – only for the
value of the variable being verified.
Ex. Verify that tan x 
sin x
.
cos x
a. Verify graphically that this is an identity.
They are.
a. Determine the non-permissible values, in degrees, for the equation.
Occurs when cos x = 0 (at 90˚ and 270˚), and can be represented as 𝜃 ≠ 90° + 180°𝑛
3
b. Verify that θ= 60˚.
L.S.
R.S.
sin 60
cos 60
tan 60
3
 3 (using exact value chart)


1
2
2
3 2

2 1
 3
LS = RS
sec 𝑥
Ex. a. Simplify the expression tan 𝑥
1
cos 𝑥
sin 𝑥
cos 𝑥
sec 𝑥
b. Determine the non-permissible values, in radians, of the variable in the expression tan 𝑥

Look at the above simplification, which expressions spend time on the denominator?
4

When does each function equal zero?

Is there a simplified expression we could use to express the non-permissible values?
𝜋
Therefore, combine the restriction is 𝑥 ≠ 2 𝑛. A restriction will occur every 90˚.
Pythagorean Identity
Recall that point P on the terminal arm of an angle θ in standard position has coordinates (cos θ,
sin θ). Consider a right triangle with a hypotenuse of 1 and legs of cos θ and sin θ.
y
P(cosθ, sinθ)
x
The hypotenuse is 1 because it is the radius of the unit circle. Apply the Pythagorean Theorem
in the right triangle to establish the Pythagorean Identity:
𝑥2 + 𝑦2 = 1
𝑐𝑜𝑠 2 𝜃 + 𝑠𝑖𝑛2 𝜃 = 1
Pythagorean Identities:
sin 2 x  cos 2 x  1
1  tan 2 x  sec2 x
5
1  cot 2 x  csc2 x
These identities can be written in several ways:
sin 2 x  1  cos 2 x
cos 2 x  1  sin 2 x
tan 2 x  sec2 x  1
cot 2 x  csc2 x  1
Ex. Verify the identity 1 + 𝑐𝑜𝑡 2 𝑥 = 𝑐𝑠𝑐 2 for the given values:
a. 𝑥 =
𝜋
6
𝜋 2
6
𝜋 2
6
L.S. = 1 + (𝑐𝑜𝑡 )
=1+
b.
R.S. = (𝑐𝑠𝑐 )
1
=
2
𝜋
(𝑡𝑎𝑛 6 )
1
𝜋 2
(𝑠𝑖𝑛 6 )
4𝜋
3
240˚(QIII) will have a ref angle of 60˚, therefore the Tan will be positive and sin will be
negative. tan 60 = √3, sin 60 =
L.S. = 1 + (𝑐𝑜𝑡
=1+
4𝜋 2
)
3
√3
2
R.S. = (𝑐𝑠𝑐
1
=
4𝜋 2
(𝑡𝑎𝑛 3 )
6
4𝜋 2
)
3
1
4𝜋 2
(𝑠𝑖𝑛 3 )
b. Use quotient identities to verify the Pythagorean identity 1 + 𝑐𝑜𝑡 2 𝑥 = 𝑐𝑠𝑐 2 x
𝑐𝑜𝑡 2 𝑥 + 1 = 𝑐𝑠𝑐 2 𝑥
𝑐𝑜𝑠 2 𝑥
1
+
1
=
𝑠𝑖𝑛2 𝑥
𝑠𝑖𝑛2 𝑥
Assignment: pg. 296-297 #1-8(odd in each), 10, 11, 13
7
LESSON 2: SUM, DIFFERENCE, AND DOUBLE-ANGLE IDENTITIES
Learning Outcomes:

Learn to apply sum, difference, and double-angle identities to verify the
equivalence of trigonometric expressions

To verify a trigonometric identity numerically and graphically using technology
Ex. Use exact values to verify the following statement:
a. sin  60  30   sin 60 cos30  sin 30 cos 60
L.S. =
R.S. =
Sum and Difference Identities:
sin( A  B)  sin A cos B  cos A sin B
sin( A  B)  sin A cos B  cos A sin B
cos( A  B)  cos A cos B  sin A sin B
cos( A  B)  cos A cos B  sin A sin B
tan A  tan B
tan( A  B) 
1  tan A tan B
tan A  tan B
tan( A  B) 
1  tan A tan B
These are on the formula sheet
Ex. Write the expression cos 88° cos 35° + sin 88° sin 35° as a single trigonometric function.
Has the same form as the right side of cos(A-B):
8
Ex. Find the exact value of sin15
break angle into exact values
sin15  sin  45  30   sin 45 cos30  sin 30 cos 45
Ex. Find the exact value of 𝑡𝑎𝑛
11𝜋
12
When faced with radians, it may be easier to switch to degrees (165˚)
Rewrite as a addition of two special angles (135 + 30)
tan 𝐴 + tan 𝐵
1 − tan 𝐴 tan 𝐵
tan 135 + tan 30
tan(135 + 30) =
1 − tan 135 tan 30
tan(𝐴 + 𝐵) =
From special triangles tan 30 =
√3
3
(tan 30 =
1
√3
, 𝑤𝑒 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙𝑖𝑧𝑒 𝑡𝑜
√3
3
For 135, it has a reference angle of 45 in QII (tan is negative). Tan 45 = 1
tan(135 + 30) =
√3
−1 + 3
√3
1 − (−1 ( 3 ))
tan(135 + 30) =
√3
−1 + 3
√3
1 − (−1 ( 3 ))
√3
−1
tan(135 + 30) = 3
√3
1+ 3
9
Simplify by combining fractions:
√3 − 3
3
tan(135 + 30) =
√3 + 3
3
tan(135 + 30) =
3
√3 − 3
×
3
√3 + 3
tan(135 + 30) =
√3 − 3
√3 + 3
Double Angle Identities:
sin 2 A  2sin A cos A
cos 2 A  cos 2 A  sin 2 A
2 tan A
tan 2 A 
1  tan 2 A
These identities are on the formula sheet
Ex. Express each in terms of a single trigonometric function:
a. 2 sin 4x cos 4x
(2 sinA cosA)
b. cos 2
1
1
A  sin 2 A
2
2
10
5
2
5
2
c. sin x cos x
Ex. Determine an identity for cos 2A that contains only the sine ratio.
cos 2𝐴 = 𝑐𝑜𝑠 2 𝐴 − 𝑠𝑖𝑛2 𝐴
Ex. Determine an identity for cos 2A that contains only the cosine ratio.
cos 2𝐴 = 𝑐𝑜𝑠 2 𝐴 − 𝑠𝑖𝑛2 𝐴
sin 2𝑥
Ex. Simplify the expression cos 2𝑥+1 to one of the three primary trigonometric ratios
=
2 sin 𝐴 cos 𝐴
(2𝑐𝑜𝑠 2 𝐴 − 1) + 1
Assignment: pg. 306-308 #1-11, 15-17, 20, 23
11
LESSON 3: PROVING IDENTITIES
Learning Outcomes:

To prove trigonometric identities algebraically

To understand the difference between verifying and proving an identity

To show that verifying the two sides of a potential identity are equal for a given value is
insufficient to prove the identity
To prove that an identity is true for all permissible values, it is necessary to express both sides of
the identity in equivalent forms. One or both sides of the identity must be algebraically
manipulated into an equivalent form to match the other side.
You cannot perform operations across the equal sign when proving a potential identity. Simplify
the expressions on each side of the identity independently.
Hints in Proving an Identity:
1. Begin with the more complex side
2. If possible, use know identities given on the formula sheet, eg. try to use
the Pythagorean identities when squares of trigonometric functions are
involved.
3. If necessary change all trigonometric ratios to sines and/or cosines, eg.
replace tan x by
sin x
1
, or sec x by
.
cos x
cos x
4. Look for factoring as a step in trying to prove an identity.
5. If there is a sum or difference of fractions, write as a single fraction
6. Occasionally, you may need to multiply the numerator or denominator of
a fraction by its conjugate.
It is usually easier to make a complicated expression simpler than it is to make a simple
expression more complicated.
12
Ex. Consider the statement
1
 cos x  sin x tan x
cos x
a. Verify the statement is true for x 

3
L.S.
R.S.
b. Prove the statement algebraically:
L.S.
R.S.
c. State the non-permissible values for x. Work in degrees.
13
2 tan 𝑥
Ex. Consider the identity 1+𝑡𝑎𝑛2 𝑥 = sin 2𝑥
a. Describe how to use a graphing calculator to verify the identity.
b. Prove the identity
L.S.
sin 𝑥
2
cos
𝑥
=
𝑠𝑒𝑐²𝑥
1
Ex. Prove that 1−sin 𝐴 =
1+sin 𝐴
𝑐𝑜𝑠2 𝐴
R.S.
= 2 sin 𝑥 cos 𝑥
is an identity for all permissible values of x.
𝐿𝑆 =
1
1 + sin 𝐴
×
1 − sin 𝐴 1 + sin 𝐴
14
2 cos 𝑥−1
1
Ex. Prove that 2𝑐𝑜𝑠2 𝑥−7 cos 𝑥+3 = cos 𝑥−3 is an identity. For what values of x is this identity
undefined?
𝐿𝑆 =
2 cos 𝑥 − 1
2𝑐𝑜𝑠 2 𝑥 − 6 cos 𝑥 − cos 𝑥 + 3
For non-permissible values, we need to find we the equations cos x – 3 = 0 or 2cos x – 1 = 0 are
undefined.
cos 𝑥 = 3
cos 𝑥 =
Assignment: pg. 314-315 #1-8, 10-15, 16-18
15
1
2
LESSON 4: SOLVING TRIGONOMETRIC EQUATIONS USING IDENTITIES
Learning Outcomes:

To solve trigonometric equations algebraically using know identities

To determine exact solutions for trigonometric equations where possible

To determine the general solution for trigonometric equations

To identify and correct errors in a solution for a trigonometric equation
Investigation
1. Graph the function 𝑦 = sin 2𝑥 − sin 𝑥 over the domain 0° < 𝑥 ≤ 360°. Describe the graph.
2. From the graph, determine an expression for the zeros of the function over the domain of all
real numbers.
3. Algebraically solve sin 2𝑥 − 𝑠𝑖𝑛𝑥 = 0 over the domain. Compare your answers to the
question above.
sin 2𝑥 − 𝑠𝑖𝑛𝑥 = 0
= 2 sin 𝑥 cos 𝑥 − sin 𝑥
sin 𝑥 = 0
1
2
Ref angle is 60˚(answers in QI and QIV)
cos 𝑥 =
16
4. Which method, graphic or algebraic, do you prefer to solve the above equation?
To solve some trigonometric equations, you need to make substitutions using the trigonometric
identities that you have studied in this chapter. This often involves ensuring that the equation is
expressed in terms of one trigonometric function.
Ex. Solve the following equation where 0  x  2 .
2 cos 2 x  3sin x  0
2(1  sin 2 x)  3sin x  0  Pythagorean identity
2  2sin 2 x  3sin x  0  Expand brackets
0  2sin 2 x  3sin x  2  Need to factor
 2sin x 1sin x  2
1
2
Positive in Quad I and II
sin x 
ref   sin 1 (0.5)  30 
x
 
, 
6 
sin x  2
No solution

6
 5

  or x  ,
6 6
6
17
Ex. Solve for x as an exact value where 0 ≤ 𝑥 ≤ 2𝜋
3 − 3 sin 𝑥 − 2𝑐𝑜𝑠 2 𝑥
3 − 3 sin 𝑥 − 2(1 − 𝑠𝑖𝑛2 𝑥) = 0
sin 𝑥 = 1
sin 𝑥 =
𝑥=
1
2
𝜋 𝜋 5𝜋
, ,
2 6 6
1
Ex. Solve the equation 𝑠𝑖𝑛2 𝑥 = 2 tan 𝑥 cos 𝑥 algebraically over the domain 0° ≤ 𝑥 < 360°.
𝑠𝑖𝑛2 𝑥 =
1 sin 𝑥
×
× cos 𝑥
2 cos 𝑥
sin 𝑥 = 0
sin 𝑥 =
18
1
2
Check for any non-permissible values: occur when cos x = 0, at 90˚, 270˚
The non-permissible values are not part of our solution set, therefore our solutions are x = 30˚,
150˚, and 180˚
b. Verify your answer algebraically.
Ex. Algebraically solve cos 2𝑥 = cos 𝑥. Give general solutions expressed in radians.
2𝑐𝑜𝑠 2 𝐴 − 1 = cos 𝑥
General Solutions:
2𝜋
3
+ 2𝜋𝑛,
19
4𝜋
3
+ 2𝜋𝑛, 0 + 2𝜋𝑛
Ex. Algebraically solve 3 cos 𝑥 + 2 = 5 sec 𝑥. Give general solutions expressed in radians.
3 cos 𝑥 + 2 = 5 sec 𝑥
3 cos 𝑥 + 2 = 5 ×
1
cos 𝑥
Only non-permissible value occurs when cosx = 0, (90˚, 270˚)
The non-permissible values do not change our original solution of 0 +2πn
Unless the domain is restricted, give general solutions. Check that the solutions for an equation
do not include non-permissible values from the original equation.
Assignment: pg. 320-321 #1-11, 14-16, 17
20