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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?
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• arcsine function
• arccosine function
• arctangent function
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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,
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Evaluate Inverse Sine Functions
Answer:
Check If
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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
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Evaluate Inverse Sine Functions
Answer:
CHECK If arcsin then sin
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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
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Find the exact value of sin–1 0.
A. 0
B.
C.
D. π
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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.
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Evaluate Inverse Cosine Functions
Answer: 0
Check If cos–1 1 = 0, then cos 0 = 1.
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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
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Evaluate Inverse Cosine Functions
CHECK If arcos
Answer:
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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
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Find the exact value of cos–1 (–1).
A.
B.
C. π
D.
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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,
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Evaluate Inverse Tangent Functions
Answer:
Check If , then tan
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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 = .
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Evaluate Inverse Tangent Functions
Answer:
Check If arctan 1 = , then tan = 1.
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Find the exact value of arctan .
A.
B.
C.
D.
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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.
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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.
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Sketch the graph of y = sin–1 2x.
A.
B.
C.
D.
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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.
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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.
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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
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Use an Inverse Trigonometric
Function
So, the viewing angle can be modeled by
Answer:
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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.
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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.
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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.
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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:
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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].
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Use Inverse Trigonometric Properties
Therefore, .
Answer:
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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.
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Find the exact value of arcsin
A.
B.
C.
D.
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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 = .
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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:
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Find the exact value of
A.
B.
C.
D.
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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
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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
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Evaluate Compositions of Trigonometric
Functions
Answer:
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Write cos(arctan x) as an algebraic expression of xthat does not involve trigonometric functions.
A.
B.
C.
D.