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### Sine and Cosine Values of Special Angles

Main definitions and formulas:

• A 45-45-90 triangle has side lengths in proportion to 1-1-√ 2.
• A 30-60-90 triangle has side lengths in proportion to 1-√ 3-2.
•  Degrees
 Cosine
 Sine
 0
 0
 1
 0
 30
 π 6
 √ 3 2
 1 2
 45
 π 4
 √ 2 2
 √ 2 2
 60
 π 3
 1 2
 √ 3 2
 90
 π 2
 0
 1
• Use these values to find sines and cosines in other quadrants. The mnemonic ASTC (All Students Take Calculus) helps you remember which ones are positive in which quadrant. (All, Sine, Tangent, Cosine)

Example 1:

Convert 120°  to radians, identify its quadrant, and find its cosine and sine.

Example 2:

Convert (5π/3)R to degrees, identify its quadrant, and find its cosine and sine.

Example 3:

Identify all angles between 0 and 2π whose sine is − [1/2], in both degrees and radians, and identify which quadrant each is in.

Example 4:

Convert 225°  to radians, identify its quadrant, and find its cosine and sine.

Example 5:

Identify all angles between 0 and 2π whose cosine is − (√3/2), in both degrees and radians, and identify which quadrant each is in.

## Convert 315° to radians, identify its quadrant, and find its cosine and sine.

• First convert 315° to radians by using the following formula: degree measure ×[(π)/(180° )] = radian measure
• 315° ×[(π)/(180° )] = [(315°π)/(180° )] = [(7π)/4]
• Locate [(7π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
• Looking at our picture, we see that [(7π)/4] is in quadrant IV. This will help us to solve for the sine and cosine values as well.
• Note that 315° is 45° from the x - axis so we can use the 45° − 45° − 90° triangle to find the sine and cosine.
• Recall that cosine is the x coordinate and sine is the y coordinate of [(7π)/4] on the unit circle
• cos [(7π)/4] = [(√2 )/2] and sin [(7π)/4] = − [(√2 )/2]
• Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
• Since [(7π)/4] is in quadrant IV we know that cosine will be positive and sine will be negative

## Convert [(4π)/3] to degrees, identify its quadrant, and find its cosine and sine.

• First convert [(4π)/3] to degrees by using the following formula: radian measure ×[(180°)/(π)] = degree measure
• [(4π)/3] ×[(180°)/(π)] = 4 ×60° = 240°
• Locate 240° on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
• Looking at our picture, we see that 240° is in quadrant III. This will help us to solve for the sine and cosine values as well.
• Note that [(4π)/3] is [(π)/3] past the x - axis so we can use the 30° − 60° − 90° triangle to find the sine and cosine.
• Recall that cosine is the x coordinate and sine is the y coordinate of 240° on the unit circle
• cos 240° = − [1/2] and sin 240° = − [(√3 )/2]
• Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
• Since 240° is in quadrant III we know that tangent will be positive so sine and cosine will be negative

## Convert 210° to radians, identify its quadrant, and find its cosine and sine.

• First convert 210° to radians by using the following formula: degree measure ×[(π)/(180°)] = radian measure
• 210° ×[(π)/(180°)] = [(210°π)/(180°)] = [(7π)/6]
• Locate [(7π)/6] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
• Looking at our picture, we see that [(7π)/6] is in quadrant III. This will help us to solve for the sine and cosine values as well.
• Note that 210° is 30° past the x - axis so we can use the 30° − 60° − 90° triangle to find the sine and cosine.
• Recall that cosine is the x coordinate and sine is the y coordinate of [(7π)/6] on the unit circle
• cos [(7π)/6] = − [(√3 )/2] and sin [(7π)/6] = − [1/2]
• Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
• Since [(7π)/6] is in quadrant III we know that tangent will be positive so cosine and sine will be negative

## Convert [(3π)/4] to degrees, identify its quadrant, and find its cosine and sine.

• First convert [(3π)/4] to degrees by using the following formula: radian measure ×[(180°)/(π)] = degree measure
• [(3π)/4] ×[(180°)/(π)] = 3 ×45° = 135°
• Locate 135° on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
• Looking at our picture, we see that 135° is in quadrant II. This will help us to solve for the sine and cosine values as well.
• Note that [(3π)/4] is [(π)/4] from the x - axis so we can use the 45° − 45° − 90° triangle to find the sine and cosine.
• Recall that cosine is the x coordinate and sine is the y coordinate of 135° on the unit circle
• cos 135° = − [(√2 )/2] and sin 135° = [(√2 )/2]
• Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
• Since 135° is in quadrant II we know that sine will be positive and cosine will be negative

## Convert 150° to radians, identify its quadrant, and find its cosine and sine.

• First convert 150° to radians by using the following formula: degree measure ×[(π)/(180°)] = radian measure
• 150° ×[(π)/(180°)] = [(150°π)/(180°)] = [(5π)/6]
• Locate [(5π)/6] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
• Looking at our picture, we see that [(5π)/6] is in quadrant II. This will help us to solve for the sine and cosine values as well.
• Note that 150° is 30° from the x - axis so we can use the 30° - 60° - 90° triangle to find the sine and cosine.
• Recall that cosine is the x coordinate and sine is the y coordinate of [(5π)/6] on the unit circle
• cos [(5π)/6] = − [(√3 )/2] and sin [(5π)/6] = [1/2]
• Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
• Since [(5π)/6] is in quadrant II we know that sine will be positive and cosine will be negative

## Evaluate sin([(3π)/4])

• Locate [(3π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine value.
• Looking at our picture, we see that [(3π)/4] is in quadrant II. This will help us to solve for the sine value.
• Note that [(3π)/4] is [(π)/4] from the x - axis so we can use the 45° − 45° − 90° triangle to find the sine value.
• Recall that sine is the y coordinate of [(3π)/4] on the unit circle
• sin [(3π)/4] = [(√2 )/2]
• Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
• Since [(3π)/4] is in quadrant II we know that sine will be positive.

## Evaluate cos 330°

• Locate 330° on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the cosine value.
• Looking at our picture, we see that 330° is in quadrant IV. This will help us to solve for the cosine value.
• Note that 330° is 30° from the x - axis so we can use the 30° - 60° - 90° triangle to find the cosine value.
• Recall that cosine is the x coordinate of 330° on the unit circle
• cos 330° = [(√3 )/2]
• Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
• Since 330° is in quadrant IV we know that cosine will be positive.

## Evaluate sin([(5π)/4])

• Locate [(5π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine value.
• Looking at our picture, we see that [(5π)/4] is in quadrant III. This will help us to solve for the sine value.
• Note that [(5π)/4] is [(π)/4] past the x - axis so we can use the 45° − 45° − 90° triangle to find the sine value.
• Recall that sine is the y coordinate of [(5π)/4] on the unit circle
• sin [(5π)/4] = − [(√2 )/2]
• Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
• Since [(5π)/4] is in quadrant III we know that sine will be negative.

## Identify all angles between 0 and 2π whose cosine is [1/2] in both degrees and radians. Identify which quadrant each is in.

• Recall that cosine is the x coordinate.
• Draw a unit circle and locate positive [1/2] along the x - axis. Draw a vertical line through that point. This will help you to identify the two points on the circle.
• Connect the points on the circle to the origin. Notice that a 30° - 60° - 90° triangle was formed.
• The first angle is 60° past 0° ⇒ 0° + 60° = 60°
• The second angle is 30° past of 270° ⇒ 270° + 30° = 300°

## Identify all angles between 0 and 2π whose sine is − [(√3 )/2] in both degrees and radians. Identify which quadrant each is in.

• Recall that sine is the y coordinate.
• Draw a unit circle and locate − [(√3 )/2] along the y - axis. Draw a horizontal line through that point. This will help you to identify the two points on the circle.
• Connect the points on the circle to the origin. Notice that a 30° - 60° - 90° triangle was formed.
• The first angle is 60° past 180° ⇒ 180° + 60° = 240°
• The second angle is 60° short of 360° ⇒ 360° − 60° = 300°

### Angle 1 Angle 2 Degree 240° 300° Radian [(4π)/3] [(5π)/3] Quadrant I IV

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.