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Trigonometry Basics (SOH-CAH-TOA) infographic - Sine, Cosine, Tangent, and Right Triangle Ratios

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Trigonometry Basics (SOH-CAH-TOA)

Sine, Cosine, Tangent, and Right Triangle Ratios

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Trigonometry helps us connect angles and side lengths in triangles, which makes it useful in geometry, physics, engineering, and navigation. The shortcut SOH CAH TOA is a memory tool for the three basic trigonometric ratios in a right triangle. Once you know one acute angle and one side, you can often find missing sides or angles. This makes right triangle problems much faster and more organized.

In a right triangle, the names opposite, adjacent, and hypotenuse depend on which acute angle you choose as θ\theta. The hypotenuse is always the longest side and lies across from the right angle. For a chosen angle θ\theta, the opposite side is directly across from θ\theta, and the adjacent side touches θ\theta but is not the hypotenuse. Using these side relationships, sine, cosine, and tangent let you build equations and solve for unknown values.

Key Facts

  • sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
  • cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
  • tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
  • SOH CAH TOA means Sine Opposite Hypotenuse, Cosine Adjacent Hypotenuse, Tangent Opposite Adjacent.
  • In a right triangle, hypotenuse2=opposite2+adjacent2\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2
  • tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Vocabulary

Right triangle
A triangle that has one angle equal to 90 degrees.
Hypotenuse
The side opposite the right angle, and it is always the longest side in a right triangle.
Opposite side
For a chosen angle θ\theta, this is the side directly across from that angle.
Adjacent side
For a chosen angle θ\theta, this is the side next to the angle that is not the hypotenuse.
Theta
A Greek letter, written as θ\theta, commonly used to represent an unknown angle.

Common Mistakes to Avoid

  • Calling the side next to θ\theta the adjacent side every time, because the hypotenuse also touches θ\theta but is never called adjacent in SOH CAH TOA problems. Always identify the hypotenuse first as the side opposite the right angle.
  • Using opposite and adjacent without choosing the reference angle, because those names change when θ\theta changes. Label sides only after deciding which acute angle is θ\theta.
  • Applying SOH CAH TOA to any triangle, because these ratios in this form are for right triangles. Check that the triangle has a 90 degree angle before using them directly.
  • Mixing up sine, cosine, and tangent in equations, because each ratio uses different side pairs. Write the full ratio from SOH CAH TOA before substituting numbers.

Practice Questions

  1. 1 In a right triangle, θ\theta is one acute angle. The side opposite θ\theta is 6 cm and the hypotenuse is 10 cm. Find sin⁡(θ)\sin(\theta) and then estimate θ\theta to the nearest degree.
  2. 2 A right triangle has an angle θ\theta with adjacent side 8 m and opposite side 6 m. Find tan⁡(θ)\tan(\theta), then use the Pythagorean theorem to find the hypotenuse.
  3. 3 A student says that if cos⁡(θ)=35\cos(\theta) = \frac{3}{5}, then the side of length 3 must always be the opposite side. Explain why this statement is incorrect using the meanings of adjacent, opposite, and hypotenuse.