Unit Circle Values | Math Coach Amy

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Key Ideas 30°, 45°, 60° Quadrantals Radians Radian axes Past 2π Sec, csc, cot Reciprocal axes

45-45-90 and 30-60-90 special right triangles

What to Remember

How to read the circle

(cos θ, sin θ) is the point on the unit circle.

Sine is the y-coordinate. Cosine is the x-coordinate.

tan θ = sin θcos θ

Tangent is undefined when cosine is 0.

30°, 45°, and 60°

The unit-circle hypotenuse is 1. Start from a 30-60-90 or 45-45-90 triangle, then attach the quadrant sign.

30°: sin = 12, cos = square root of 32, tan = square root of 33

45°: sin = square root of 22, cos = square root of 22, tan = 1

60°: sin = square root of 32, cos = 12, tan = square root of 3

Quadrant I: all three are positive. Quadrant II: sine. Quadrant III: tangent. Quadrant IV: cosine.

Quadrantal angles

These angles land on the axes.

0° and 360°: (1, 0)

90° and π2: (0, 1)

180° and π: (−1, 0)

270° and 3π2: (0, −1)

2π is the same point as 0°.

Past one full turn, subtract 360° or 2π. The value stays the same.

Flip for secant, cosecant, and cotangent

csc θ = 1sin θ

sec θ = 1cos θ

cot θ = 1tan θ

First find sine, cosine, or tangent. Then flip it. The sign stays the same.

If the denominator is 0, the value is undefined.

Easy 1

30°, 45°, 60° reference angles

Degrees from 0° to 360°. Sine, cosine, and tangent. Reference angles are 30°, 45°, and 60°.

Example

Find the exact value of sin 150°.

Reference angle: 180° − 150° = 30°.

sin θ = opp/hyp = 1/2

y-values are positive in the second quadrant, so sine is positive in quadrant 2.

sin 150° = 12

Answer: 12

Find the exact value of cos 30°.

Show Answer

cos θ = adj/hyp = square root of 3/2

Answer: square root of 32

Find the exact value of sin 45°.

Show Answer

sin θ = opp/hyp = 1/square root of 2 = square root of 22

Answer: square root of 22

Find the exact value of tan 60°.

Show Answer

tan θ = opp/adj = square root of 3/1 = square root of 3

Answer: square root of 3

Find the exact value of sin 60°.

Show Answer

sin θ = opp/hyp = square root of 3/2

Answer: square root of 32

Find the exact value of sin 120°.

Show Answer

Reference angle: 180° − 120° = 60°.

sin θ = opp/hyp = square root of 3/2

y-values are positive in the second quadrant, so sine is positive in quadrant 2.

sin 120° = square root of 32

Answer: square root of 32

Find the exact value of cos 135°.

Show Answer

Reference angle: 180° − 135° = 45°.

cos θ = adj/hyp = 1/square root of 2 = square root of 22

x-values are negative in the second quadrant, so cosine is negative in quadrant 2.

cos 135° = −square root of 22

Answer: −square root of 22

Find the exact value of tan 135°.

Show Answer

Reference angle: 180° − 135° = 45°.

tan θ = opp/adj = 1/1 = 1

y-values are positive and x-values are negative in the second quadrant, so tangent is negative in quadrant 2.

tan 135° = −1

Answer: −1

Find the exact value of tan 150°.

Show Answer

Reference angle: 180° − 150° = 30°.

tan θ = opp/adj = 1/square root of 3 = square root of 33

y-values are positive and x-values are negative in the second quadrant, so tangent is negative in quadrant 2.

tan 150° = −square root of 33

Answer: −square root of 33

Find the exact value of cos 210°.

Show Answer

Reference angle: 210° − 180° = 30°.

cos θ = adj/hyp = square root of 3/2

x-values are negative in the third quadrant, so cosine is negative in quadrant 3.

cos 210° = −square root of 32

Answer: −square root of 32

Find the exact value of tan 210°.

Show Answer

Reference angle: 210° − 180° = 30°.

tan θ = opp/adj = 1/square root of 3 = square root of 33

y-values and x-values are negative in the third quadrant, so tangent is positive in quadrant 3.

tan 210° = square root of 33

Answer: square root of 33

Find the exact value of sin 225°.

Show Answer

Reference angle: 225° − 180° = 45°.

sin θ = opp/hyp = 1/square root of 2 = square root of 22

y-values are negative in the third quadrant, so sine is negative in quadrant 3.

sin 225° = −square root of 22

Answer: −square root of 22

Find the exact value of cos 240°.

Show Answer

Reference angle: 240° − 180° = 60°.

cos θ = adj/hyp = 1/2

x-values are negative in the third quadrant, so cosine is negative in quadrant 3.

cos 240° = −12

Answer: −12

Find the exact value of cos 315°.

Show Answer

Reference angle: 360° − 315° = 45°.

cos θ = adj/hyp = 1/square root of 2 = square root of 22

x-values are positive in the fourth quadrant, so cosine is positive in quadrant 4.

cos 315° = square root of 22

Answer: square root of 22

Find the exact value of sin 330°.

Show Answer

Reference angle: 360° − 330° = 30°.

sin θ = opp/hyp = 1/2

y-values are negative in the fourth quadrant, so sine is negative in quadrant 4.

sin 330° = −12

Answer: −12

Find the exact value of tan 300°.

Show Answer

Reference angle: 360° − 300° = 60°.

tan θ = opp/adj = square root of 3/1 = square root of 3

y-values are negative and x-values are positive in the fourth quadrant, so tangent is negative in quadrant 4.

tan 300° = −square root of 3

Answer: −square root of 3

Easy 2

Quadrantals

Degrees from 0° to 360° on the x-axis or the y-axis. Sine, cosine, and tangent.

Example

Find the exact value of tan 90°.

90° is on the positive y-axis.

The point is (0, 1).

tan θ = sin θcos θ

sin θ = 1

cos θ = 0

The denominator is 0.

tan 90° is undefined.

Answer: undefined

Find the exact value of sin 0°.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

sin θ = y

y = 0

sin 0° = 0

Answer: 0

Find the exact value of cos 0°.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

cos θ = x

x = 1

cos 0° = 1

Answer: 1

Find the exact value of tan 0°.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

tan θ = sin θcos θ

sin θ = 0

cos θ = 1

tan 0° = 01

tan 0° = 0

Answer: 0

Find the exact value of sin 90°.

Show Answer

90° is on the positive y-axis.

The point is (0, 1).

sin θ = y

y = 1

sin 90° = 1

Answer: 1

Find the exact value of cos 90°.

Show Answer

90° is on the positive y-axis.

The point is (0, 1).

cos θ = x

x = 0

cos 90° = 0

Answer: 0

Find the exact value of sin 180°.

Show Answer

180° is on the negative x-axis.

The point is (−1, 0).

sin θ = y

y = 0

sin 180° = 0

Answer: 0

Find the exact value of cos 180°.

Show Answer

180° is on the negative x-axis.

The point is (−1, 0).

cos θ = x

x = −1

cos 180° = −1

Answer: −1

Find the exact value of tan 180°.

Show Answer

180° is on the negative x-axis.

The point is (−1, 0).

tan θ = sin θcos θ

sin θ = 0

cos θ = −1

tan 180° = 0−1

tan 180° = 0

Answer: 0

Find the exact value of sin 270°.

Show Answer

270° is on the negative y-axis.

The point is (0, −1).

sin θ = y

y = −1

sin 270° = −1

Answer: −1

Find the exact value of cos 270°.

Show Answer

270° is on the negative y-axis.

The point is (0, −1).

cos θ = x

x = 0

cos 270° = 0

Answer: 0

Find the exact value of tan 270°.

Show Answer

270° is on the negative y-axis.

The point is (0, −1).

tan θ = sin θcos θ

sin θ = −1

cos θ = 0

The denominator is 0.

tan 270° is undefined.

Answer: undefined

Find the exact value of sin 360°.

Show Answer

360° − 360° = 0°

0° is on the positive x-axis.

The point is (1, 0).

sin θ = y

y = 0

sin 360° = 0

Answer: 0

Find the exact value of cos 360°.

Show Answer

360° − 360° = 0°

0° is on the positive x-axis.

The point is (1, 0).

cos θ = x

x = 1

cos 360° = 1

Answer: 1

Find the exact value of tan 360°.

Show Answer

360° − 360° = 0°

0° is on the positive x-axis.

The point is (1, 0).

tan θ = sin θcos θ

sin θ = 0

cos θ = 1

tan 360° = 01

tan 360° = 0

Answer: 0

Medium 1

Reference angles in radians

Radians from 0 to 2π. Same kind of reference angles as Easy 1. Sine, cosine, and tangent.

Example

Find the exact value of tan 2π3.

Reference angle: π − 2π3 = 3π3 − 2π3

3π3 − 2π3 = π3.

π3 is 60°.

tan θ = opp/adj = square root of 3/1 = square root of 3

y-values are positive and x-values are negative in the second quadrant, so tangent is negative in quadrant 2.

tan 2π3 = −square root of 3

Answer: −square root of 3

Find the exact value of sin π6.

Show Answer

π6 is 30°.

sin θ = opp/hyp = 1/2

sin π6 = 12

Answer: 12

Find the exact value of cos π4.

Show Answer

π4 is 45°.

cos θ = adj/hyp = 1/square root of 2 = square root of 22

cos π4 = square root of 22

Answer: square root of 22

Find the exact value of tan π4.

Show Answer

π4 is 45°.

tan θ = opp/adj = 1/1 = 1

tan π4 = 1

Answer: 1

Find the exact value of cos π3.

Show Answer

π3 is 60°.

cos θ = adj/hyp = 1/2

cos π3 = 12

Answer: 12

Find the exact value of sin 7π4.

Show Answer

Reference angle: 2π − 7π4 = 8π4 − 7π4

8π4 − 7π4 = π4.

π4 is 45°.

sin θ = opp/hyp = 1/square root of 2 = square root of 22

y-values are negative in the fourth quadrant, so sine is negative in quadrant 4.

sin 7π4 = −square root of 22

Answer: −square root of 22

Find the exact value of cos 5π6.

Show Answer

Reference angle: π − 5π6 = 6π6 − 5π6

6π6 − 5π6 = π6.

π6 is 30°.

cos θ = adj/hyp = square root of 3/2

x-values are negative in the second quadrant, so cosine is negative in quadrant 2.

cos 5π6 = −square root of 32

Answer: −square root of 32

Find the exact value of sin 3π4.

Show Answer

Reference angle: π − 3π4 = 4π4 − 3π4

4π4 − 3π4 = π4.

π4 is 45°.

sin θ = opp/hyp = 1/square root of 2 = square root of 22

y-values are positive in the second quadrant, so sine is positive in quadrant 2.

sin 3π4 = square root of 22

Answer: square root of 22

Find the exact value of cos 2π3.

Show Answer

Reference angle: π − 2π3 = 3π3 − 2π3

3π3 − 2π3 = π3.

π3 is 60°.

cos θ = adj/hyp = 1/2

x-values are negative in the second quadrant, so cosine is negative in quadrant 2.

cos 2π3 = −12

Answer: −12

Find the exact value of sin 7π6.

Show Answer

Reference angle: 7π6 − π = 7π6 − 6π6

7π6 − 6π6 = π6.

π6 is 30°.

sin θ = opp/hyp = 1/2

y-values are negative in the third quadrant, so sine is negative in quadrant 3.

sin 7π6 = −12

Answer: −12

Find the exact value of tan 5π4.

Show Answer

Reference angle: 5π4 − π = 5π4 − 4π4

5π4 − 4π4 = π4.

π4 is 45°.

tan θ = opp/adj = 1/1 = 1

y-values and x-values are negative in the third quadrant, so tangent is positive in quadrant 3.

tan 5π4 = 1

Answer: 1

Find the exact value of sin 4π3.

Show Answer

Reference angle: 4π3 − π = 4π3 − 3π3

4π3 − 3π3 = π3.

π3 is 60°.

sin θ = opp/hyp = square root of 3/2

y-values are negative in the third quadrant, so sine is negative in quadrant 3.

sin 4π3 = −square root of 32

Answer: −square root of 32

Find the exact value of tan 4π3.

Show Answer

Reference angle: 4π3 − π = 4π3 − 3π3

4π3 − 3π3 = π3.

π3 is 60°.

tan θ = opp/adj = square root of 3/1 = square root of 3

y-values and x-values are negative in the third quadrant, so tangent is positive in quadrant 3.

tan 4π3 = square root of 3

Answer: square root of 3

Find the exact value of tan 7π4.

Show Answer

Reference angle: 2π − 7π4 = 8π4 − 7π4

8π4 − 7π4 = π4.

π4 is 45°.

tan θ = opp/adj = 1/1 = 1

y-values are negative and x-values are positive in the fourth quadrant, so tangent is negative in quadrant 4.

tan 7π4 = −1

Answer: −1

Find the exact value of tan 11π6.

Show Answer

Reference angle: 2π − 11π6 = 12π6 − 11π6

12π6 − 11π6 = π6.

π6 is 30°.

tan θ = opp/adj = 1/square root of 3 = square root of 33

y-values are negative and x-values are positive in the fourth quadrant, so tangent is negative in quadrant 4.

tan 11π6 = −square root of 33

Answer: −square root of 33

Find the exact value of cos 5π3.

Show Answer

Reference angle: 2π − 5π3 = 6π3 − 5π3

6π3 − 5π3 = π3.

π3 is 60°.

cos θ = adj/hyp = 1/2

x-values are positive in the fourth quadrant, so cosine is positive in quadrant 4.

cos 5π3 = 12

Answer: 12

Medium 2

Quadrantals

Radians from 0 to 2π on the axes. Sine, cosine, and tangent.

Example

Find the exact value of cos π.

π is 180°.

180° is on the negative x-axis.

The point is (−1, 0).

cos θ = x

x = −1

cos π = −1

Answer: −1

Find the exact value of sin 0.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

sin θ = y

y = 0

sin 0 = 0

Answer: 0

Find the exact value of cos 0.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

cos θ = x

x = 1

cos 0 = 1

Answer: 1

Find the exact value of tan 0.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

tan θ = sin θcos θ

sin θ = 0

cos θ = 1

tan 0 = 01

tan 0 = 0

Answer: 0

Find the exact value of sin π2.

Show Answer

π2 is 90°.

90° is on the positive y-axis.

The point is (0, 1).

sin θ = y

y = 1

sin π2 = 1

Answer: 1

Find the exact value of cos π2.

Show Answer

π2 is 90°.

90° is on the positive y-axis.

The point is (0, 1).

cos θ = x

x = 0

cos π2 = 0

Answer: 0

Find the exact value of tan π2.

Show Answer

π2 is 90°.

90° is on the positive y-axis.

The point is (0, 1).

tan θ = sin θcos θ

sin θ = 1

cos θ = 0

The denominator is 0.

tan π2 is undefined.

Answer: undefined

Find the exact value of sin π.

Show Answer

π is 180°.

180° is on the negative x-axis.

The point is (−1, 0).

sin θ = y

y = 0

sin π = 0

Answer: 0

Find the exact value of tan π.

Show Answer

π is 180°.

180° is on the negative x-axis.

The point is (−1, 0).

tan θ = sin θcos θ

sin θ = 0

cos θ = −1

tan π = 0−1

tan π = 0

Answer: 0

Find the exact value of sin 3π2.

Show Answer

3π2 is 270°.

270° is on the negative y-axis.

The point is (0, −1).

sin θ = y

y = −1

sin 3π2 = −1

Answer: −1

Find the exact value of cos 3π2.

Show Answer

3π2 is 270°.

270° is on the negative y-axis.

The point is (0, −1).

cos θ = x

x = 0

cos 3π2 = 0

Answer: 0

Find the exact value of tan 3π2.

Show Answer

3π2 is 270°.

270° is on the negative y-axis.

The point is (0, −1).

tan θ = sin θcos θ

sin θ = −1

cos θ = 0

The denominator is 0.

tan 3π2 is undefined.

Answer: undefined

Find the exact value of sin 2π.

Show Answer

2π − 2π = 0

0 is 0°.

0° is on the positive x-axis.

The point is (1, 0).

sin θ = y

y = 0

sin 2π = 0

Answer: 0

Find the exact value of cos 2π.

Show Answer

2π − 2π = 0

0 is 0°.

0° is on the positive x-axis.

The point is (1, 0).

cos θ = x

x = 1

cos 2π = 1

Answer: 1

Find the exact value of tan 2π.

Show Answer

2π − 2π = 0

0 is 0°.

0° is on the positive x-axis.

The point is (1, 0).

tan θ = sin θcos θ

sin θ = 0

cos θ = 1

tan 2π = 01

tan 2π = 0

Answer: 0

Hard 1

More than 2π

The angle is greater than 2π. Subtract full turns, then use a value you know.

Example

Find the exact value of sin 13π6.

13π6 − 2π = 13π6 − 12π6

13π6 − 12π6 = π6

π6 is 30°.

sin θ = opp/hyp = 1/2

sin 13π6 = 12

Answer: 12

Find the exact value of cos 7π3.

Show Answer

7π3 − 2π = 7π3 − 6π3

7π3 − 6π3 = π3

π3 is 60°.

cos θ = adj/hyp = 1/2

cos 7π3 = 12

Answer: 12

Find the exact value of tan 9π4.

Show Answer

9π4 − 2π = 9π4 − 8π4

9π4 − 8π4 = π4

π4 is 45°.

tan θ = opp/adj = 1/1 = 1

tan 9π4 = 1

Answer: 1

Find the exact value of sin 5π2.

Show Answer

5π2 − 2π = 5π2 − 4π2

5π2 − 4π2 = π2

π2 is 90°.

90° is on the positive y-axis.

The point is (0, 1).

sin θ = y

y = 1

sin 5π2 = 1

Answer: 1

Find the exact value of sin 11π4.

Show Answer

11π4 − 2π = 11π4 − 8π4

11π4 − 8π4 = 3π4

Reference angle: π − 3π4 = 4π4 − 3π4

4π4 − 3π4 = π4.

π4 is 45°.

sin θ = opp/hyp = 1/square root of 2 = square root of 22

y-values are positive in the second quadrant, so sine is positive in quadrant 2.

sin 11π4 = square root of 22

Answer: square root of 22

Find the exact value of cos 8π3.

Show Answer

8π3 − 2π = 8π3 − 6π3

8π3 − 6π3 = 2π3

Reference angle: π − 2π3 = 3π3 − 2π3

3π3 − 2π3 = π3.

π3 is 60°.

cos θ = adj/hyp = 1/2

x-values are negative in the second quadrant, so cosine is negative in quadrant 2.

cos 8π3 = −12

Answer: −12

Find the exact value of sin 17π6.

Show Answer

17π6 − 2π = 17π6 − 12π6

17π6 − 12π6 = 5π6

Reference angle: π − 5π6 = 6π6 − 5π6

6π6 − 5π6 = π6.

π6 is 30°.

sin θ = opp/hyp = 1/2

y-values are positive in the second quadrant, so sine is positive in quadrant 2.

sin 17π6 = 12

Answer: 12

Find the exact value of cos 3π.

Show Answer

3π − 2π = π

π is 180°.

180° is on the negative x-axis.

The point is (−1, 0).

cos θ = x

x = −1

cos 3π = −1

Answer: −1

Find the exact value of tan 7π2.

Show Answer

7π2 − 2π = 7π2 − 4π2

7π2 − 4π2 = 3π2

3π2 is 270°.

270° is on the negative y-axis.

The point is (0, −1).

tan θ = sin θcos θ

sin θ = −1

cos θ = 0

The denominator is 0.

tan 7π2 is undefined.

Answer: undefined

Find the exact value of tan 23π6.

Show Answer

23π6 − 2π = 23π6 − 12π6

23π6 − 12π6 = 11π6

Reference angle: 2π − 11π6 = 12π6 − 11π6

12π6 − 11π6 = π6.

π6 is 30°.

tan θ = opp/adj = 1/square root of 3 = square root of 33

y-values are negative and x-values are positive in the fourth quadrant, so tangent is negative in quadrant 4.

tan 23π6 = −square root of 33

Answer: −square root of 33

Find the exact value of cos 25π6.

Show Answer

25π6 − 4π = 25π6 − 24π6

25π6 − 24π6 = π6

π6 is 30°.

cos θ = adj/hyp = square root of 3/2

cos 25π6 = square root of 32

Answer: square root of 32

Hard 2

Secant, cosecant, and cotangent

30°, 45°, and 60°. First find sine, cosine, or tangent. Then flip it.

Example

Find the exact value of sec 30°.

sec θ = 1cos θ

cos θ = adj/hyp = square root of 3/2

Flip it.

sec 30° = 2square root of 3

sec 30° = 2square root of 33

Answer: 2square root of 33

Find the exact value of csc 30°.

Show Answer

csc θ = 1sin θ

sin θ = opp/hyp = 1/2

Flip it.

csc 30° = 2

Answer: 2

Find the exact value of cot 30°.

Show Answer

cot θ = 1tan θ

tan θ = opp/adj = 1/square root of 3 = square root of 33

Flip it.

cot 30° = 3square root of 3

cot 30° = square root of 3

Answer: square root of 3

Find the exact value of sec 45°.

Show Answer

sec θ = 1cos θ

cos θ = adj/hyp = 1/square root of 2 = square root of 22

Flip it.

sec 45° = 2square root of 2

sec 45° = square root of 2

Answer: square root of 2

Find the exact value of csc 45°.

Show Answer

csc θ = 1sin θ

sin θ = opp/hyp = 1/square root of 2 = square root of 22

Flip it.

csc 45° = 2square root of 2

csc 45° = square root of 2

Answer: square root of 2

Find the exact value of cot 45°.

Show Answer

cot θ = 1tan θ

tan θ = opp/adj = 1/1 = 1

Flip it.

cot 45° = 1

Answer: 1

Find the exact value of sec 60°.

Show Answer

sec θ = 1cos θ

cos θ = adj/hyp = 1/2

Flip it.

sec 60° = 2

Answer: 2

Find the exact value of csc 60°.

Show Answer

csc θ = 1sin θ

sin θ = opp/hyp = square root of 3/2

Flip it.

csc 60° = 2square root of 3

csc 60° = 2square root of 33

Answer: 2square root of 33

Find the exact value of cot 60°.

Show Answer

cot θ = 1tan θ

tan θ = opp/adj = square root of 3/1 = square root of 3

Flip it.

cot 60° = 1square root of 3

cot 60° = square root of 33

Answer: square root of 33

Hard 3

Quadrantal reciprocals

Secant, cosecant, and cotangent on the axes. First find sine, cosine, or tangent. Then flip it.

Example

Find the exact value of cot 90°.

90° is on the positive y-axis.

The point is (0, 1).

cot θ = 1tan θ

tan θ is undefined.

cot θ = cos θsin θ

cos θ = 0

sin θ = 1

cot 90° = 01

cot 90° = 0

Answer: 0

Find the exact value of sec 0°.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

sec θ = 1cos θ

cos θ = x

x = 1

Flip it.

sec 0° = 1

Answer: 1

Find the exact value of csc 0°.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

csc θ = 1sin θ

sin θ = y

y = 0

The denominator is 0.

csc 0° is undefined.

Answer: undefined

Find the exact value of cot 0°.

Show Answer

0° is on the positive x-axis.

The point is (1, 0).

cot θ = 1tan θ

tan θ = sin θcos θ

sin θ = 0

cos θ = 1

tan θ = 01

tan θ = 0

The denominator is 0.

cot 0° is undefined.

Answer: undefined

Find the exact value of sec 90°.

Show Answer

90° is on the positive y-axis.

The point is (0, 1).

sec θ = 1cos θ

cos θ = x

x = 0

The denominator is 0.

sec 90° is undefined.

Answer: undefined

Find the exact value of csc 90°.

Show Answer

90° is on the positive y-axis.

The point is (0, 1).

csc θ = 1sin θ

sin θ = y

y = 1

Flip it.

csc 90° = 1

Answer: 1

Find the exact value of sec 180°.

Show Answer

180° is on the negative x-axis.

The point is (−1, 0).

sec θ = 1cos θ

cos θ = x

x = −1

Flip it.

sec 180° = −1

Answer: −1

Find the exact value of csc 180°.

Show Answer

180° is on the negative x-axis.

The point is (−1, 0).

csc θ = 1sin θ

sin θ = y

y = 0

The denominator is 0.

csc 180° is undefined.

Answer: undefined

Find the exact value of cot 180°.

Show Answer

180° is on the negative x-axis.

The point is (−1, 0).

cot θ = 1tan θ

tan θ = sin θcos θ

sin θ = 0

cos θ = −1

tan θ = 0−1

tan θ = 0

The denominator is 0.

cot 180° is undefined.

Answer: undefined

Find the exact value of sec 270°.

Show Answer

270° is on the negative y-axis.

The point is (0, −1).

sec θ = 1cos θ

cos θ = x

x = 0

The denominator is 0.

sec 270° is undefined.

Answer: undefined

Find the exact value of csc 270°.

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270° is on the negative y-axis.

The point is (0, −1).

csc θ = 1sin θ

sin θ = y

y = −1

Flip it.

csc 270° = −1

Answer: −1

Find the exact value of cot 270°.

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270° is on the negative y-axis.

The point is (0, −1).

cot θ = 1tan θ

tan θ is undefined.

cot θ = cos θsin θ

cos θ = 0

sin θ = −1

cot 270° = 0−1

cot 270° = 0

Answer: 0

Find the exact value of sec 360°.

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360° − 360° = 0°

0° is on the positive x-axis.

The point is (1, 0).

sec θ = 1cos θ

cos θ = x

x = 1

Flip it.

sec 360° = 1

Answer: 1

Find the exact value of csc 360°.

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360° − 360° = 0°

0° is on the positive x-axis.

The point is (1, 0).

csc θ = 1sin θ

sin θ = y

y = 0

The denominator is 0.

csc 360° is undefined.

Answer: undefined

Find the exact value of cot 360°.

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360° − 360° = 0°

0° is on the positive x-axis.

The point is (1, 0).

cot θ = 1tan θ

tan θ = sin θcos θ

sin θ = 0

cos θ = 1

tan θ = 01

tan θ = 0

The denominator is 0.

cot 360° is undefined.

Answer: undefined

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