Tangent is the ratio of an angles sine value to its cosine value.
C
c
The sine value of an angle is the y-coordinate on the unit circle.
D
d
The sine value of an angle is the y-coordinate on the unit circle.
E
e
Tangent is the ratio of the sine value to the cosine value.
F
f
Subtract 2π from the radian measure to find the corresponding angle on the first lap around the unit circle.
G
g
For which angles are the sine value and cosine value equal?
H
h
For which angles are the sine value and cosine value opposite numbers?
A
a
cos 3π/4=- 1/sqrt(2)
B
b
tan 4π/3=sqrt(3)
C
c
sin 11π/6=- 1/2
D
d
sin 3π/4=1/sqrt(2)
E
e
tan 5π/4=1
F
f
tan 17π/6=- sqrt(3)/3
G
g
θ=π/4, θ=5π/4
H
h
θ=3π/4, θ=7π/4
Practice makes perfect
a
To answer these questions, we can use the following diagram.
Since the cosine of a trig expression is given by the horizontal axis on the unit circle, we can determine the exact value by identifying the x-value of the angle of rotation that corresponds to 3π4 radians.
From the diagram, we see that cos 3π4=- 1sqrt(2).
b
The tangent value of an angle is the ratio of the angles sine value to the cosine value.
tan θ = sin θ/cos θTherefore, we have to identify both the sine and cosine values when the angle of rotation is 4π3.
With this information, we can find the exact tangent value.
Let's identify 11π6 radians on the unit circle. In this case we are looking for the angles sine value. As already explained, this is the y-value on the unit circle.
From the diagram, we see that sin 11π6=- 12.
d
Similar to Part A, we have an angle of rotation of 3π4. However, in this case we are looking for the sine value.
From the diagram, we see that sin 3π4= 1sqrt(2).
e
Like in Part B, we have to identify both the since value and cosine value when the angle of rotation is 5π4.
With this information, we can find the exact tangent value.
A measure of 2π radians equals one lap around the unit circle, and 4π equals two laps around the unit circle. Therefore, an angle of 17π6 radians must describe an angle on the second lap around the unit circle. Because of the periodicity of a tangent curve, we can find the corresponding angle on the first lap by subtracting 2π from the given radian measure.
If we find tan 5π6 we will also determine tan 17π6. Like in Parts B and E, we have to identify both the sine and cosine value when the angle of rotation is 5π6.
With this information, we can find the exact tangent value.
As already explained, the tangent of an angle is the ratio of the angle's sine value to the angle's cosine value.
tan θ = sin θ/cos θThe only way that tan θ can equal 1 is if the corresponding sine and cosine value are the same. We have two possibilities to choose from.
As we can see, when the angle of rotation is π4 or 5π4, the sine and cosine value are the same. Therefore, the tangent value must be 1.
tan π/4 &= .1 /sqrt(2)./.1 /sqrt(2).=1
tan 5π/4 &= - .1 /sqrt(2)./- .1 /sqrt(2).=1
We have two solutions, θ = π4 and θ = 5π4.
h
Similar to Part G, we have to find the sine and cosine values that produce a quotient of - 1. The only way this can happen is if the cosine and sine values are opposite numbers. For the given interval, there are two possibilities.
As we can see, when the angle of rotation is 3π4 or 7π4, the sine and cosine value are opposite numbers. Therefore, the tangent value must be - 1.
tan 3π/4 &= - .1 /sqrt(2)./.1 /sqrt(2).=- 1
tan 7π/4 &= .1 /sqrt(2)./- .1 /sqrt(2).=- 1
We have two solutions, θ = 3π4 and θ = 7π4.
