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Inverse Trig Functions
Objective: Evaluate the Inverse Trig Functions
Existence of Inverse Functions
• We know that for a function to have an inverse, it must be one-to-one, that is, must pass the horizontal line test. Neither graph below passes the horizontal line test so they don’t have an inverse. In order for y = sinx or y = cosx to have an inverse, we must restrict the domain.
Inverse Sine
• Look at the sine graph. If we take just a part of the graph, that part will pass the horizontal line test, and will have an inverse.
• The domain of this graph is –/2 < x < /2• The range of this graph is -1 < y < 1
Inverse Cosine
• Look at the cosine graph. If we take just a part of the graph, that part will pass the horizontal line test, and will have an inverse.
• The domain of this graph is 0 < x < • The range of this graph is -1 < y < 1
Definition of Inverse Sine
• The inverse sine function is defined by:
or if and only if
The domain of this inverse is -1 < x < 1And the range of the inverse is –/2 < y < /2
xy 1sin xy arcsin
xy sin
Definition of Inverse Cosine
• The inverse cosine function is defined by:
or if and only if
The domain of this inverse is -1 < x < 1And the range of the inverse is 0 < y <
xy 1cosxy arccos
xy cos
Graphs of Inverse Functions
• Take a look at the graphs of the inverse sine and cosine functions. Notice the domain and range and how they are the opposite of the original functions.
Domain Restrictions
• What the domain restrictions mean to us is that when we talk about a positive sine value, we are in the first quadrant. When we talk about a negative sine value, we are in the fourth quadrant.
• When we talk about a positive cosine value, we again are in the first quadrant, but when we talk about a negative cosine value, we are in the second quadrant.
Example 1
• Find the exact value of the following:
2
1cos 1
2
1sin 1
2
3arcsin
Example 1
• Find the exact value of the following:
2
1cos 1
2
1sin 1
2
3arcsin
6
3
3
You Try
• Find the exact value of the following:
2
2cos 1
2
1sin 1
2
2arcsin
You Try
• Find the exact value of the following:
2
2cos 1
2
1sin 1
2
2arcsin
6
4
34
You Try
• Find the exact value of the following:
)0(cos 1)1(sin 1 )1arccos(
You Try
• Find the exact value of the following:
)0(cos 1)1(sin 1 )1arccos(
2
2
Inverse Tangent
• Lets look at the graph of the tangent. There are many breaks in the graph, but the graph does not pass the horizontal line test. If we just look at one part of the graph, it will pass the horizontal line test.
• We will restrict the domain to –/2 < x < /2• The range will be all numbers.
Inverse Tangent
• Lets look at the graph of the inverse tangent. Again, we switch the domain and the range. Like the inverse sine, the inverse tangent is defined in the first and fourth quadrant only.
• Now the range is –/2 < x < /2• The domain will be all numbers.
Inverse Tangent
• Find the exact value of the following:
)0(tan 1)1(tan 1)3(tan 1
Inverse Tangent
• Find the exact value of the following:
)0(tan 1)1(tan 1)3(tan 1
4
0
3
You Try
• Find the exact value of the following:
)1(tan 1
3
3tan 1 )3(tan 1
You Try
• Find the exact value of the following:
)1(tan 1
3
3tan 1 )3(tan 1
6
4
3
Inverse Functions
• We are also able to find inverse values of other angles by using our calculator.
• Find the exact value in degrees of the following:
)79(.sin 1)89(.cos 1 )31.6(tan 1
Inverse Functions
• We are also able to find inverse values of other angles by using our calculator.
• Find the exact value in degrees of the following:
)79(.sin 1)89(.cos 1 )31.6(tan 1
013.27 018.52 099.80
You Try
• We are also able to find inverse values of other angles by using our calculator.
• Find the exact value in degrees of the following:
)54.(sin 1 )33.(cos 1 )21.4(tan 1
You Try
• We are also able to find inverse values of other angles by using our calculator.
• Find the exact value in degrees of the following:
)54.(sin 1 )33.(cos 1 )21.4(tan 1
027.109 068.32 064.76
Inverse Functions
• Given find the exact value of all 6 trig
functions.
5
4sin 1
Inverse Functions
• Given find the exact value of all 6 trig
functions.• Remember, this means that the sin = 4/5. This tells
us that y = 4 and r = 5. Using the Pythagorean Theorem, x = 3. We know we are in the first quadrant since the value is positive.
5
4sin
5
4sin 1
4
5csc
5
3cos
3
5sec
3
4tan
4
3cot
Example 6
• Find the exact value of:
3
2arccostan
5
3sincos 1
Example 6
• Find the exact value of:
• The first one tells us that cos = 2/3, so x = 2 and r = 3 so y = . The tan =
3
2arccostan
5
3sincos 1
5 .2/5
Example 6
• Find the exact value of:
• The first one tells us that cos = 2/3, so x = 2 and r = 3 so y = . The tan =
• The second one tells us that y = -3 and r = 5 so x = 4. The cos = 4/5.
3
2arccostan
5
3sincos 1
5 .2/5
You Try
• Find the exact value of:
5
12arctansin
25
7costan 1
You Try
• Find the exact value of:
5
12arctansin
25
7costan 1
13
12sin
13,5,12
rxy
7
24tan
24,25,7
yrx
Class work
• Pages 511
• 2-14 even
• 38, 50
Homework
• Pages 511-512
• 1-15 odd
• 19-33 odd (in degrees)
• 37, 39, 49, 51, 53
