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You found and graphed the inverses of relations and functions. (Lesson 1-7)
LEQ: How do we evaluate and graph inverse trigonometric functions & find compositions of trigonometric functions?
• arcsine function
• arccosine function
• arctangent function
Evaluate Inverse Sine Functions
A. Find the exact value of , if it exists.
Find a point on the unit circle on the interval
with a y-coordinate of . When t =
Therefore,
Evaluate Inverse Sine Functions
Answer:
Check If
Evaluate Inverse Sine Functions
B. Find the exact value of , if it exists.
Find a point on the unit circle on the interval
with a y-coordinate of When t = , sin t =
Therefore, arcsin
Evaluate Inverse Sine Functions
Answer:
CHECK If arcsin then sin
Evaluate Inverse Sine Functions
C. Find the exact value of sin–1 (–2π), if it exists.
Because the domain of the inverse sine function is [–1, 1] and –2π < –1, there is no angle with a sine of –2π. Therefore, the value of sin–1(–2π) does not exist.
Answer: does not exist
Find the exact value of sin–1 0.
A. 0
B.
C.
D. π
Evaluate Inverse Cosine Functions
A. Find the exact value of cos–11, if it exists.
Find a point on the unit circle on the interval [0, π] with an x-coordinate of 1. When t = 0, cos t = 1. Therefore, cos–11 = 0.
Evaluate Inverse Cosine Functions
Answer: 0
Check If cos–1 1 = 0, then cos 0 = 1.
Evaluate Inverse Cosine Functions
B. Find the exact value of , if it exists.
Find a point on the unit circle on the interval [0, π] with
an x-coordinate of When t =
Therefore, arccos
Evaluate Inverse Cosine Functions
CHECK If arcos
Answer:
Evaluate Inverse Cosine Functions
C. Find the exact value of cos–1(–2), if it exists.
Since the domain of the inverse cosine function is [–1, 1] and –2 < –1, there is no angle with a cosine of –2. Therefore, the value of cos–1(–2) does not exist.
Answer: does not exist
Find the exact value of cos–1 (–1).
A.
B.
C. π
D.
Evaluate Inverse Tangent Functions
A. Find the exact value of , if it exists.
Find a point on the unit circle on the interval
such that When t = ,
tan t = Therefore,
Evaluate Inverse Tangent Functions
Answer:
Check If , then tan
Evaluate Inverse Tangent Functions
B. Find the exact value of arctan 1, if it exists.
Find a point on the unit circle in the interval
such that When t = , tan t =
Therefore, arctan 1 = .
Evaluate Inverse Tangent Functions
Answer:
Check If arctan 1 = , then tan = 1.
Find the exact value of arctan .
A.
B.
C.
D.
Sketch Graphs of Inverse Trigonometric
Functions
Sketch the graph of y = arctan
By definition, y = arctan and tan y = are equivalent on
for < y < , so their graphs are the same. Rewrite
tan y = as x = 2 tan y and assign values to y on the
interval to make a table to values.
Answer:
Sketch Graphs of Inverse Trigonometric
Functions
Then plot the points (x, y) and connect them with a smooth curve. Notice that this curve is contained within its asymptotes.
Sketch the graph of y = sin–1 2x.
A.
B.
C.
D.
Use an Inverse Trigonometric
Function
A. MOVIES In a movie theater, a 32-foot-tall screen is located 8 feet above ground. Write a function modeling the viewing angle θ for a person in the theater whose eye-level when sitting is 6 feet above ground.
Draw a diagram to find the measure of the viewing angle. Let θ1 represent the angle formed from eye-level to the bottom of the screen, and let θ2 represent the angle formed from eye-level to the top of the screen.
Use an Inverse Trigonometric
Function
So, the viewing angle is θ = θ2 – θ1. You can use the tangent function to find θ1 and θ2. Because the eye-level of a seated person is 6 feet above the ground, the distance opposite θ1 is 8 – 6 feet or 2 feet long.
Use an Inverse Trigonometric
Function
opp = 2 and adj = d
Inverse tangent function
The distance opposite θ2 is (32 + 8) – 6 feet or 34 feet
Inverse tangent function
opp = 34 and adj = d
Use an Inverse Trigonometric
Function
So, the viewing angle can be modeled by
Answer:
Use an Inverse Trigonometric
Function
B. MOVIES In a movie theater, a 32-foot-tall screen is located 8 feet above ground-level. Determine the distance that corresponds to the maximum viewing angle.
The distance at which the maximum viewing angle occurs is the maximum point on the graph. You can use a graphing calculator to find this point.
Answer: about 8.2 ft
Use an Inverse Trigonometric
Function
From the graph, you can see that the maximum viewing angle occurs approximately 8.2 feet from the screen.
MATH COMPETITION In a classroom, a 4 foot tall screen is located 6 feet above the floor. Write a function modeling the viewing angle θ for a student in the classroom whose eye-level when sitting is 3 feet above the floor.
A.
B.
C.
D.
Use Inverse Trigonometric Properties
A. Find the exact value of , if it exists.
Therefore,
The inverse property applies because lies on the interval [–1, 1].
Answer:
Use Inverse Trigonometric Properties
B. Find the exact value of , if it exists.
Notice that does not lie on the interval [0, π].
However, is coterminal with – 2π or which
is on the interval [0, π].
Use Inverse Trigonometric Properties
Therefore, .
Answer:
Use Inverse Trigonometric Properties
Answer: does not exist
C. Find the exact value of , if it exists.
Because tan x is not defined when x = ,
arctan does not exist.
Find the exact value of arcsin
A.
B.
C.
D.
Evaluate Compositions of Trigonometric
Functions
Find the exact value of
Because the cosine function is positive in Quadrants I and IV, and the domain of the inverse cosine function is restricted to Quadrants I and II, u must lie in Quadrant I.
To simplify the expression, let u = cos–1
so cos u = .
Evaluate Compositions of Trigonometric
Functions
Using the Pythagorean Theorem, you can find that the length of the side opposite is 3. Now, solve for sin u.
opp = 3 and hyp = 5
Sine function
So,
Answer:
Find the exact value of
A.
B.
C.
D.
Write cot (arccos x) as an algebraic expression of x that does not involve trigonometric functions.
Let u = arcos x, so cos u = x.
Because the domain of the inverse cosine function is restricted to Quadrants I and II, u must lie in Quadrant I or II. The solution is similar for each quadrant, so we will solve for Quadrant I.
Evaluate Compositions of Trigonometric
Functions
Evaluate Compositions of Trigonometric
Functions
From the Pythagorean Theorem, you can find that the
length of the side opposite to u is . Now, solve
for cot u.
Cotangent function
So, cot(arcos x) = .
opp = and adj = x
Evaluate Compositions of Trigonometric
Functions
Answer:
Write cos(arctan x) as an algebraic expression of xthat does not involve trigonometric functions.
A.
B.
C.
D.
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