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 cosine of an angle is the x-coordinate on the unit circle.
A
a
x=2Ï€/3, x=4Ï€/3
B
b
x=Ï€/6, x=7Ï€/6
C
c
x=0, x=Ï€
D
d
x=Ï€/4, x=7Ï€/4
Practice makes perfect
a
To find the angles that makes the equation true, we can use the following diagram.
The cosine of a trig expression is given by the unit circle's horizontal axis. Examining the diagram, we notice that two angles result in a cosine value of - 12.
Our two solutions are θ= 2π3 and θ= 4π3.
b
The tangent value of an angle is the ratio of the angle's sine value to its cosine value.
tan θ = sin θ/cos θTherefore, we have to identify an angle of rotation where the ratio of the sine value to the cosine value equals sqrt(3)3. Notice that this is the same ratio as 1sqrt(3). We have two such angles.
If we calculate the ratio of the angle's sine value to their cosine value, we can see that the ratio equals 1sqrt(3).
tan π/6 &= .1 /2./.sqrt(3) /2. = 1/sqrt(3)
tan 7Ï€/6 &= -.1 /2./-.sqrt(3) /2. = 1/sqrt(3)
Our two solutions are x= π6 and x= 7π6.
c
The sine of a trig expression is given by the unit circle's vertical axis. Examining the diagram, we notice that two angles result in a sine value of 0.
There are two angles within the given interval that give the desired sine value, x=0 and x=Ï€.
d
Since we are working with a cosine value, we have to find the angle of rotation that corresponds to a value of sqrt(2)2 on the horizontal axis. Notice that sqrt(2)2 is the same thing as 1sqrt(2). We have two angles.
There are two angles within the given interval that give the desired cosine value, x= π4 and x= 7π4.
