Download Section 5.1 - Fundamental Identities HW: # 1, 2, 4, 23

Section 5.1 - Fundamental Identities
HW: # 1, 2, 4, 23 - 33, 41, 42, 46, 51, 67, 68
Recall the following identities that we covered in section 1.5.
1
csc θ
cos θ = sec1 θ
tan θ = cot1 θ
sin θ =
csc θ =
sec θ =
cot θ =
1
sin θ
1
cos θ
1
tan θ
sin2 θ + cos2 θ = 1
tan θ =
1 + tan2 θ = sec2 θ
1 + cot2 θ = csc2 θ
cot θ =
sin θ
cos θ
cos θ
sin θ
Consider the following picture:
Theorem: We have just shown that sin(−θ) = − sin θ and cos(−θ) = cos θ. Recall that
we saw this in problems 49 and 50 of section 1.3.
Example: Use the above results for sine and cosine to find a similar theorem for tangent.
Example: (#38) Simplify sec θ cot θ sin θ.
Example: (#47) Simplify tan θ + cot θ.
Example: (#49) Simplify sin θ(csc θ − sin θ).
Example: (#48) Simplify (sec θ + csc θ)(cos θ − sin θ).
Example: (#50) Simplify
1 + tan2 θ
.
1 + cot2 θ
Example: (#52) Simplify
tan(−θ)
.
sec θ
Section 5.3 - Sum and Difference Identities for Cosine
HW: # 3, 7, 10, 33, 45, 49, 63
Theorem:
cos(A − B) = cos A cos B + sin A sin B
cos(A + B) = cos A cos B − sin A sin B
Example: Use the calculator to verify that the formulas work for A = 23◦ and B = 17◦ .
Example: Find the exact value of cos 15◦ .
Example: Find the exact value of cos( 11π
12 ).
Example: Prove the cofunction identity cos(90◦ − θ) = sin θ using the cosine difference
identity. (Note: To prove an identity, start on one side of the equal sign and work your
way to the other side. Do not work on both sides at the same time.)
Question: If we were using radian measure, what would the previous problem look like?
Example: (#35) Use the cofunction identities to find an angle θ that makes the statement
sec θ = csc( 2θ + 20◦ ) true.
Example: (#48) Find the exact values of cos(s + t) and cos(s − t) if sin s =
sin t = − 13 , s in quadrant II and t in quadrant IV.
2
3
and
Section 5.4 - Sum and Difference Identities for Sine and Tangent
HW: # 10, 13, 15, 48, 55, 56
Theorem:
sin(A − B) = sin A cos B − cos A sin B
sin(A + B) = sin A cos B + cos 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
Example: Find the exact value of tan(−105◦ ).
Example: (#61) Verify that the statement
tan x − tan y
sin(x − y)
=
is an identity.
sin(x + y)
tan x + tan y
Section 5.5 - Double-Angle Identities
HW: # 11, 17, 22, 25, 32, 63
Theorem:
cos 2A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A
sin 2A = 2 sin A cos A
2 tan A
tan 2A =
1 − tan2 A
Example: Let’s prove the cosine and sine formulas.
Example: (#18) If cos 2X = .3, then what is 2(sin X)2 ?
Example: (#12) Find the values of all six trigonometric functions for the angle 2β given
that cos β = − 12
13 and sin β > 0.
◦
◦
Example: Use an identity to write the expression 2 sin 22 12 cos 22 12 as a single number.
Example: (#26) Do the same for the expression cos2
π
8
− 12 .
Example: (#51) Verify the identity (sin γ + cos γ)2 = sin 2γ + 1.
Example: (#57) Verify the identity sin 4α = 4 sin α cos α cos 2α.
Example: (#69) Express the function cos 3x as a trigonometric function of x.
Section 6.1 - Inverse Trigonometric Functions
HW: # 6, 14, 16, 22, 35, 38, 40, 41, 43, 46, 60, 61, 79
Suppose sin x = 0. What is x?
Definition: The inverse sine function or arcsine is y = sin−1 x = arcsin x if x = sin y
where −1 ≤ x ≤ 1 and − π2 ≤ y ≤ π2 .
In other words, y = sin−1 x means “what angle y has a sine of x and lives between − π2
and π2 ?”
The same sort of definition applies to the other trigonometric functions. All we need to
keep straight is where the inverse trigonometric functions live.
Arcsine and arccosecant live on the interval [− π2 , π2 ]. Arctangent lives on the interval
(− π2 , π2 ). Arccosine and arcsecant live on the interval [0, π]. Arccotangent lives on the
interval (0, π).
Problem: Why do we have some of the intervals including the endpoints and some of the
intervals not including the endpoints?
Notice that the ranges of the inverse trigonometric functions are chosen so that the trigonometric functions are positive in one quadrant and negative in one quadrant.
Example:
√ Find each exactly in radians.
a. sin−1 ( 2/2)
b. tan−1 (−1)
√
c. arccsc(−2/ 3)
d. arccos(−1)
Example: Use the calculator to find each of the following.
a. y = sin−1 (.1234) where y is in radians
b. y = csc−1 (1.713) where y is in degrees
c. y = arcsin(1.34) where y is in degrees
d. y = sec−1 (1.41) where y is in radians
Example: Find the exact value of each of the following. If the answer is an angle, use
radian measure.
a. sec(sin−1 (3/5))
b. sin(sin−1 (1/2))
c. sin−1 (sin π)
d. tan−1 (sin(π/2))
Example: Find the exact value of tan[arcsin(3/5) + arccos(−3/5)].
Section 6.2 - Trigonometric Equations
HW: # 10, 11, 17, 25, 27, 29
Example: (#8) Solve the equation sin x + 2 = 3 in the interval [0, 2π).
Example: (#12) Solve the equation sec2 x + 2 = −1 in the interval [0, 2π).
Example: (#18) Solve the equation tan3 x = 3 tan x in the interval [0, 2π).
Example: (#15) Solve the equation cos2 x + 2 cos x + 1 = 0 in the interval [0, 2π).
Example: (#54) Find all solutions, in radians, of the equation sin2 x − sin x = 0.
Question: What if the previous problem had asked for answers in degrees? (Note that
we could also have used 0 + πn.)
Example: (#26) Solve the equation cos2 θ = sin2 θ + 1 in the interval [0◦ , 360◦ ).