Trigonometry

Sine, Cosine and Tangent Explained (SOHCAHTOA and the Unit Circle)

By Math Solving Space · · 2 min read

On this page
  1. SOH CAH TOA
  2. Exact values to remember
  3. The unit circle
  4. Signs in each quadrant (ASTC)
  5. Worked example: sin, cos and tan of 150°
  6. Worked example: 300°
  7. Common mistakes
  8. Practice questions
  9. Frequently asked questions

In a right triangle, sine is opposite ÷ hypotenuse, cosine is adjacent ÷ hypotenuse, and tangent is opposite ÷ adjacent, remembered as SOH CAH TOA. On the unit circle, the point at angle θ\theta has coordinates (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta), which extends trigonometry to every angle.

SOH CAH TOA

For an acute angle θ\theta in a right triangle:

sin⁡θ=opphypcos⁡θ=adjhyptan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}

Example: a ladder 5 m long leans at 60° to the ground. How high does it reach?

height=5sin⁡60∘=5×32≈4.33 m\text{height} = 5\sin 60^\circ = 5 \times \frac{\sqrt{3}}{2} \approx 4.33 \text{ m}

Exact values to remember

θ\theta sin⁡θ\sin\theta cos⁡θ\cos\theta tan⁡θ\tan\theta
0∘0^\circ 00 11 00
30∘30^\circ 12\frac{1}{2} 32\frac{\sqrt{3}}{2} 33\frac{\sqrt{3}}{3}
45∘45^\circ 22\frac{\sqrt{2}}{2} 22\frac{\sqrt{2}}{2} 11
60∘60^\circ 32\frac{\sqrt{3}}{2} 12\frac{1}{2} 3\sqrt{3}
90∘90^\circ 11 00 undefined

The unit circle

Draw a circle of radius 1 centred at the origin. Turn anticlockwise by θ\theta from the positive x-axis. The point you land on is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta). This works for any angle, including those above 90° and negative angles.

It also gives the key identity, from Pythagoras:

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

Signs in each quadrant (ASTC)

Quadrant Angles Positive
1 0°–90° All
2 90°–180° Sin
3 180°–270° Tan
4 270°–360° Cos

Worked example: sin, cos and tan of 150°

  1. 150° is in quadrant 2.
  2. The reference angle (to the x-axis) is 180∘−150∘=30∘180^\circ - 150^\circ = 30^\circ.
  3. Only sine is positive in quadrant 2:
sin⁡150∘=12,cos⁡150∘=−32,tan⁡150∘=−33\sin 150^\circ = \frac{1}{2}, \quad \cos 150^\circ = -\frac{\sqrt{3}}{2}, \quad \tan 150^\circ = -\frac{\sqrt{3}}{3}

Worked example: 300°

Quadrant 4, reference angle 60∘60^\circ, only cosine positive: sin⁡300∘=−32\sin 300^\circ = -\frac{\sqrt{3}}{2} and cos⁡300∘=12\cos 300^\circ = \frac{1}{2}.

Common mistakes

  • Calculator in the wrong mode (degrees vs radians).
  • Mixing up opposite and adjacent: they depend on which angle you are using.
  • Forgetting the sign in quadrants 2–4.

Practice questions

  1. Find cos⁡45∘\cos 45^\circ exactly.
  2. Find sin⁡210∘\sin 210^\circ exactly.
  3. Find tan⁡120∘\tan 120^\circ exactly.

Answers: 1) 22\frac{\sqrt{2}}{2} 2) −12-\frac{1}{2} 3) −3-\sqrt{3}

The trigonometry calculator gives exact values with the reference angle and an animated unit circle. To solve whole triangles use the triangle calculator, and read degrees and radians to switch units.

Frequently asked questions

Why is tan 90° undefined?

tan θ = sin θ ÷ cos θ, and cos 90° = 0, so it would mean dividing by zero.

What are csc, sec and cot?

They are the reciprocals: csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ.

Can sin θ be bigger than 1?

No, for real angles. On the unit circle the y-coordinate is always between −1 and 1.

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