5.1 -- Fundamental Identities
Wednesday, August 07, 2013
1:04 PM
Note: We will also use alternative forms of the fundamental identities.
For example, 2 other forms of sin2 ϴ + cos2 Ө = 1 are
and
Ex1: Finding Trig Function Values Given 1 Value and the Quadrant
If
and θ is in QII, find each function value.
1) sec θ
2) sin θ
3) cot ( θ)
Classroom Ex1: If
1) sin θ
and θ is in QIV, find each function value.
2) tan θ
3) sec ( θ)
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** to avoid a common error, when taking the square root, be sure to choose the sign based on the quadrant of θ and the
function being evaluated.
Ex2: Writing 1 Trig Function in Terms of Another
Write cos x in terms of tan x.
• Find a trig value that is related to both cos x and tan x.
• Then we will substitute so that the equation only has cos x and tan x and then solve for cos x.
Classroom Ex2: Write tan θ in terms of cos θ.
Ex3: Rewriting an Expression in Terms of Sine and Cosine
Write
in terms of sin θ and cos θ, then simplify the expression so that no quotients appear.
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Classroom Ex3: Write
that no quotients appear.
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in terms of sin θ and cos θ, then simplify the expression so
5.2 -- Verifying Trig Identities
Wednesday, August 07, 2013
1:04 PM
** note -- verifying is NOT the same as solving equations!
* it is important to become familiar with this section of verifying trig identities!!!
Hints for Verifying Trig Identities
1. Learn the fundamental identities given in section 5.1. Whenever you see either side of an identity,
the other side should come to mind. Also, be aware of equivalent forms of the identities. For ex,
is an alternative form of
2. Try to rewrite the more complicated side of the equation so that it is identical to the simpler side.
3. It is sometimes helpful to express all trig functions in the equation in terms of sin and cos and then
simplify the results.
4. Usually any factoring or indicated algebraic operations should be performed. The algebraic
identities are often used in verifying trig identities.
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
a2 - b2 = (a + b)(a - b)
a3 - b3 = (a - b)(a2 + ab + b2)
a3 + b3 = (a + b)(a2 - ab + b2)
The sum or diff of 2 trig expressions can be found in the same way as any other rational expressions
5. As you select substitutions, keep in mind the side you are not changing, because it represents your goal. For
ex, to verify the identity try to think of an identity that relates tan x to cos x. In this case secant is the best link.
6. If a rational expression contains 1 + sin x, multiplying the numerator and denominator by 1 - sin x would
give you 1 - sin2 x, which could be replaced by cos2 x. Similar procedures apply for 1 - sin x, 1 + cos x, and 1 cos x.
Ex1: Verifying an Identity (Working with One Side)
Verify that the following equation is an identity
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Classroom Ex1: Verify that the following is an identity.
Ex2: Verify that the following is an identity.
Classroom Ex2: Verify that the following is an identity.
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Ex3:Verify that the following is an identity.
Classroom Ex3: Verify that the following is an identity.
Ex4:Verify that the following is an identity.
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Classroom Ex4: Verify that the following is an identity.
If both sides of an identity appear to be equally complex, the identity can be verified by working
independently on the left and right side until each side is changed into some common 3rd side.
Ex5: Verifying an Identity (Working with Both Sides)
Verify that the following equation is an identity
Classroom Ex5: Verifying an Identity (Working with Both Sides)
Verify that the following equation is an identity
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** you will need to know how to
factor x2 + 2xy + y2 into (x + y)2
You can also start by trying to just
use sine and cosine throughout
Quiz review
Tuesday, June 16, 2015
8:48 AM
Chapter 5 Page 8
5.3 -- Sum and Difference Identities for Cosine
Wednesday, August 07, 2013
cos (A
1:04 PM
B) does not equal cos A
cos B
You cannot distribute into the angles
The Sum Identity and Difference Identity for Cosine are as follows
• These identities are important in calculus and useful in certain applications. For example,
the method shown in Ex1 below can be applied to find the exact value for cos 15⁰. If the
angles are given in radians it is generally easier to convert them to degrees.
Ex1: Finding Exact Cosine Function Values
Find the exact value of each expression.
1) cos 15⁰
2) cos
** unit circle values will be used -- get familiar again!!!
Classroom Ex1: Find the exact value of each expression.
1) Cos ( 75⁰)
2) cos
Chapter 5 Page 9
3) cos 87⁰ cos 93⁰ ̶ sin 87⁰ sin 93⁰
3) cos 173⁰ cos 83⁰ + sin 173⁰ sin 83⁰
Cofunction Identities
cos (90⁰ - θ) = sin θ
sin (90⁰ - θ) = cos θ
tan (90⁰ - θ) = cot θ
cot (90⁰ - θ) = tan θ
sec (90⁰ - θ) = csc θ
csc (90⁰ - θ) = sec θ
COFUNCTIONS
sine and cosine
tangent and cotangent
secant and cosecant
The angles of cofunctions are ALWAYS complementary and therefore their sum is 90⁰ or 2π radians.
Ex2:Using Cofunction Identities to Find θ
Find one value of θ or x that satisfies each of the following
2) sin θ = cos( 30⁰)
1) cot θ = tan 25⁰
3)
Classroom Ex2
1) sec θ = csc 62⁰
3)
2) tan θ = cot( 54⁰)
• If either angle A or B is a quadrantal angle then the sum and difference identities can be
written as a single function of A or B.
Ex3: Reducing cos(A - B) to a Function of a Single Variable
Write cos (180⁰ θ) as a trig function of θ alone.
Classroom Ex3: Write cos(90⁰ + θ) as a trig function of θ alone.
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Ex4: Finding cos(s + t) Given Information about s and t
Suppose that
, and both s and t are in QII. Find cos(s + t).
Classroom Ex4:
Suppose that
, and both s and are in QIV. Find cos(s
Ex6: Verifying an Identity
Verify that the following equation is an identity.
** make the left side look identical to the
right side by changing sec into cos and
then using the cosine difference identity
Classroom Ex6: Verify that the following equation is an identity.
Chapter 5 Page 11
t).
5.4 -- Sum and Difference Identities for Sine and
Tangent
Wednesday, August 07, 2013
1:05 PM
Sine of a Sum and Difference
Tangent of a Sum or Difference
Sin(A + B) = sin A cos B + cos A sin B
Sin(A B) = sin A cos B cos A Sin B
Ex1: Finding Exact Sine and Tangent Function Values
Find the exact value of each expression. Changing radians to degrees makes it easier to solve.
1) sin 75⁰
2)
3) sin 40⁰ cos 160⁰ cos 40⁰ sin 160⁰
Classroom Ex1:
Find the exact value of each expression.
1) sin ( 15°)
2)
Ex2: Writing Functions as Expressions Involving Functions of θ
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3)
Ex2: Writing Functions as Expressions Involving Functions of θ
Write each function as an expression involving functions of θ
1) Sin (30° + θ)
2) tan (45° θ)
3) sin (180°
θ)
Classroom Ex2:Write each function as an expression involving functions of θ
1) sin (120° + θ)
2) tan (θ + 3π)
3) sin (θ 270°)
Ex3: Finding Function Values and the Quadrant of A + B
Suppose that A and B are angles in the standard position, with
and
Find each of the following.
1) sin (A + B)
2) tan (A + B)
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3) the quadrant of A + B
Classroom Ex3:
Suppose that A and B are angles in the standard position, with
and
1) sin (A
Find each of the following.
B)
2) tan (A
B)
Ex4: Verifying an Identity Using Sum and Difference Identities
Verify that the equation is an identity.
Classroom Ex4: Verify that the equation is an identity.
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3) the quadrant of A
B
Quiz Review?
Thursday, August 22, 2013
11:18 AM
Chapter 5 Page 15
5.5 -- Double Angle Identities
Wednesday, August 07, 2013
1:05 PM
Double-Angle Identities occur when A = B in the identities for the sum of 2 angles. To derive an
expression for cos 2A, we let B = A in the identity
(** if you want to see the derivation,
it is in the book on pp. 224-225)
cos(A + B) = cos A cos B sin A sin B.
** note: cos 2A ≠ 2 cos A
Ex1: Finding Function Values of 2θ Given Information about θ
Given
Classroom Ex1: Given
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Ex2: Finding Function Values of θ Given Information about 2θ
Find the values of the 6 trig functions of θ if cos 2θ = and
Classroom Ex2:
Find the values of the 6 trig functions of θ if cos 2θ =
Ex3: Verifying a Double-Angle Identity
Verify that the following equation is an identity:
Chapter 5 Page 17
and
Classroom Ex3: Verify that the following equation is an identity:
Ex4: Simplifying Expressions Using Double-Angle Identities
Simplify each expression.
1) cos2 7x - sin2 7x
2) sin 15° cos 15°
Classroom Ex4: Simplify each expression.
1) 2 cos2 5x - 1
Ex5: Deriving a Multiple-Angle Identity
Write sin 3x in terms of sin x.
Chapter 5 Page 18
2) sin 165° cos 165°
Classroom Ex5: Write cos 3x in terms of cos x.
Looking ahead to calculus -- the product-to-sum identities are used in calculus to find
integrals of functions that are products of trig functions.
Ex7: Using a Product-to-Sum Identity
Write
as the sum or difference of 2 functions.
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Classroom Ex7: Write
as the sum or difference of 2 functions.
**you will need to use the negative-angle identities from section 5.1
Ex8: Using a Sum-to-Product Identity
Write sin 2θ sin 4θ as a product of 2 functions.
Classroom Ex8: Write cos 3θ + cos 7θ as a product of 2 functions.
Chapter 5 Page 20
5.6 -- Half-Angle Identities
Wednesday, August 07, 2013
1:07 PM
In the half-angle identities, the
symbol ± indicates that the sign
is chosen based on the function
under consideration and the
quadrant of
Ex1: Using a Half-Angle Identity to Find an Exact Value
Find the exact value of cos 15° using the half-angle identity for cosine.
Classroom Ex1: Find the exact value of sin 22.5° using the half-angle identity for sine.
Chapter 5 Page 21
Ex2: Using a Half-Angle Identity to Find an Exact Value
Find the exactly value of tan 22.5° using the identity
Classroom Ex2: Find the exactly value of tan 75° using the identity
Ex3: Finding Function Values of
Given
Classroom Ex3: Given
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Given Information about s.
Classroom Ex3: Given
Ex4: Simplifying Expressions Using the Half-Angle Identities
Simplify each expression.
2)
1)
Classroom Ex4:
2)
1)
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Ex5: Verifying an Identifying
Verify that the following equation is an identity.
Classroom Ex5: Verify that the following equation is an identity.
Chapter 5 Page 24
Test Review?
Friday, August 23, 2013
1:04 PM
Chapter 5 Page 25