Consider the angle θ = - 150 ^(∘) . We want to sketch it in standard position. An angle is in standard position if its vertex is located at the origin and one of the rays lays on the positive x-axis. This ray is called the initial side of the angle.
The second ray — the position of the initial ray after rotating it θ degrees — is called the terminal side. Since our angle is negative, we have to rotate the initial side clockwise.
The initial side and the terminal side, along with the vertex at the origin, form the angle θ in the standard position.
We want to find a coterminal angle to θ, which measure is in the interval [ 0 ^(∘), 360 ^(∘) ). Two angles are coterminal if they share the same initial and terminal side. Let's label this angle α.
We want to find α, which measure is in the interval [ 0 ^(∘), 360 ^(∘) ). Note that if the angle α is coterminal to the angle θ, the measure of α is equal to the measure of θ plus a multiple of 360 ^(∘).
α = θ + n * 360 ^(∘)
This identity holds for any integer value of n. Let's substitute some integer values of n and find the possible measures of α.
n
α = θ + n * 360 ^(∘)
α
- 2
- 150 ^(∘) + (- 2) * 360 ^(∘)
- 870 ^(∘)
- 1
- 150 ^(∘) + (- 1) * 360 ^(∘)
- 510 ^(∘)
0
- 150 ^(∘) + 0 * 360 ^(∘)
- 150 ^(∘)
1
- 150 ^(∘) + 1 * 360 ^(∘)
210 ^(∘)
2
- 150 ^(∘) + 2 * 360 ^(∘)
570 ^(∘)
The only value of α in the interval of [0 ^(∘), 360 ^(∘)) is 210 ^(∘).
Now, we want to convert the angle measure of θ = - 150 ^(∘) into radians. To convert degrees to radians, we multiply the angle measure in degrees by π180 ^(∘) because π radians = 180 ^(∘). Let's do it!
We want to find the reference angle θ'. The reference angle is the smallest angle made by the terminal side of the angle θ and the x-axis. It is always an acute angle or a 90^(∘) angle.
Our angle θ has a negative measure. This is why we will use the coterminal angle α to find the reference angle. Let's see the relationship between the coterminal angle α and the reference angle θ' in the picture!
Next, we can calculate the measure of the reference angle in degrees and radians simultaneously!
Angle θ
Coterminal Angle α
Reference Angle θ '
- 150^(∘)
- 150^(∘) +360^(∘) = 210^(∘)
210^(∘)-180^(∘) = 30^(∘)
-5π/6
-5π/6 + 2π = 7π/6
7π/6 - π= π/6
The measure of the reference angle θ ' is 30 ^(∘) or π6.
We want to find the values of the six trigonometric functions of θ. Let's list them!
sin θ, cos θ, tan θ,
csc θ, sec θ, cot θ
We know that the absolute values of the trigonometric functions for θ and θ ' are the same. The values might differ in sign. We can use the reference angle θ' and decide on the sign of the value depending on the quadrant in which θ lies. First, let's find sin θ ', cos θ', and tan θ'. Since θ ' = 30 ^(∘), we know these values.
Function
Value
sin 30 ^(∘)
1/2
cos 30 ^(∘)
sqrt(3)/2
tan 30 ^(∘)
sqrt(3)/2
We can see that angle θ lies in Quadrant III.
Recall the signs of sin θ, cos θ, and tan θ based on the quadrant θ lies in.
Since θ lies in Quadrant III, sin θ and cos θ are negative, and tan θ is positive.
Function of θ '
Value
Function of θ
Value
sin 30 ^(∘)
1/2
sin( - 150 ^(∘)) = - sin 30 ^(∘)
- 1/2
cos 30 ^(∘)
sqrt(3)/2
cos( - 150 ^(∘)) = - cos 30 ^(∘)
- sqrt(3)/2
tan 30 ^(∘)
sqrt(3)/3
tan( - 150 ^(∘)) = tan 30 ^(∘)
sqrt(3)/3
We found the values of sin θ, cos θ, and tan θ. Note that csc θ, sec θ, and cot θ are the inverses of these values.
Identity
Value
csc θ = 1/sin θ
- 2
sec θ = 1/cos θ
- 2/sqrt(3)
cot θ = 1/tan θ
3/sqrt(3)
Let's summarize our findings in a table.
Function
Value
sin (-150^(∘))
- 1/2
cos (-150^(∘))
- sqrt(3)/2
tan (-150^(∘))
sqrt(3)/3
csc (-150^(∘))
- 2
sec (-150^(∘))
- 2/sqrt(3)
cot (-150^(∘))
3/sqrt(3)
We can also rationalize denominators of the secant and cosecant. This means multiplying both the numerator and denominator by sqrt(3).
