Trigonometric Ratios: Sin, Cos and Tan Explained

Trigonometry begins with three simple ratios: sine, cosine and tangent. Once you understand where they come from and how to choose the right one, you can find any missing side or angle in a right-angled triangle in a few lines of working. This guide explains the ratios clearly and walks you through worked examples step by step.

What Are Trigonometric Ratios?

A trigonometric ratio compares two sides of a right-angled triangle relative to one of its acute angles, usually called \( \theta \). The three sides are named from the point of view of that angle:

• Hypotenuse (H): the longest side, opposite the right angle.
• Opposite (O): the side directly across from \( \theta \).
• Adjacent (A): the side next to \( \theta \) that is not the hypotenuse.

The labels "opposite" and "adjacent" change if you switch to the other acute angle, but the hypotenuse never changes. Always label the triangle before you do anything else.

The Three Basic Ratios

The definitions are:

$$ \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \qquad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \qquad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} $$

A popular memory aid is SOH CAH TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

These ratios depend only on the angle, not on the size of the triangle. Two similar triangles with the same angle \( \theta \) give exactly the same value of \( \sin\theta \), which is why calculators can store these values.

A Useful Connection

Dividing sine by cosine gives tangent:

$$ \frac{\sin\theta}{\cos\theta} = \frac{O/H}{A/H} = \frac{O}{A} = \tan\theta $$

You will use this fact again and again in later topics such as identities and equations.

Exact Values for Special Angles

Some angles appear so often in Matric, FSc and O Level papers that you should know their exact values without a calculator.

Notice that \( \sin 30^\circ = \cos 60^\circ \) and \( \sin 60^\circ = \cos 30^\circ \). In general, \( \sin\theta = \cos(90^\circ - \theta) \), because the opposite side of one acute angle is the adjacent side of the other.

Worked Example 1: Finding All Three Ratios

Question: In a right-angled triangle, the side opposite \( \theta \) is 5 cm and the adjacent side is 12 cm. Find \( \sin\theta \), \( \cos\theta \) and \( \tan\theta \).

Step 1: Find the hypotenuse using Pythagoras.

$$ H = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm} $$

Step 2: Apply each ratio.

$$ \sin\theta = \frac{5}{13}, \qquad \cos\theta = \frac{12}{13}, \qquad \tan\theta = \frac{5}{12} $$

Step 3: Check. \( \dfrac{\sin\theta}{\cos\theta} = \dfrac{5/13}{12/13} = \dfrac{5}{12} = \tan\theta \). The answers are consistent.

Worked Example 2: Finding a Missing Side

Question: A ladder 10 m long leans against a wall, making an angle of \( 35^\circ \) with the ground. How high up the wall does it reach, and how far is its foot from the wall?

Step 1: Label. The ladder is the hypotenuse, \( H = 10 \). The height on the wall is opposite the \( 35^\circ \) angle. The distance along the ground is adjacent.

Step 2: Height (O and H known, so use sine).

$$ \sin 35^\circ = \frac{O}{10} \quad\Rightarrow\quad O = 10 \sin 35^\circ = 10 \times 0.5736 = 5.74 \text{ m} $$

Step 3: Distance from the wall (A and H, so use cosine).

$$ \cos 35^\circ = \frac{A}{10} \quad\Rightarrow\quad A = 10 \cos 35^\circ = 10 \times 0.8192 = 8.19 \text{ m} $$

Step 4: Check with Pythagoras. \( 5.736^2 + 8.192^2 = 32.90 + 67.11 = 100.01 \approx 10^2 \). Correct.

Worked Example 3: Finding a Missing Angle

Question: A right-angled triangle has opposite side 7 cm and adjacent side 4 cm. Find \( \theta \) to two decimal places.

Step 1: Choose the ratio. We know O and A, so use tangent.

$$ \tan\theta = \frac{7}{4} = 1.75 $$

Step 2: Use the inverse function.

$$ \theta = \tan^{-1}(1.75) = 60.26^\circ $$

The inverse functions \( \sin^{-1} \), \( \cos^{-1} \) and \( \tan^{-1} \) take a ratio and return the angle. They are written as "shift + sin" and so on on most calculators.

How to Choose the Right Ratio

Follow this quick routine for any right-triangle problem:

1. Mark the angle you are working with.
2. Label H, O and A.
3. Circle the two sides involved: the one you know and the one you want.
4. Pick the ratio that contains exactly those two sides.
5. Rearrange and solve.

Common Mistakes

• Calculator in the wrong mode. If your answer for \( \sin 30^\circ \) is not 0.5, your calculator is in radian mode. Switch to degrees.
• Mixing up opposite and adjacent. These depend on which angle you use. Re-label if the question changes the angle.
• Using SOH CAH TOA on a non-right triangle. These ratios only work directly when there is a right angle. For other triangles use the sine and cosine rules.
• Rounding too early. Keep full calculator values until the last step to avoid errors creeping in.
• Writing \( \sin^{-1} x \) as \( 1/\sin x \). The inverse sine is an angle, not a reciprocal.

Key Takeaways

• Sine, cosine and tangent compare pairs of sides relative to an angle in a right-angled triangle.
• SOH CAH TOA tells you which sides each ratio uses.
• Learn the exact values for 30°, 45° and 60°; they appear constantly in exams.
• Use inverse functions to find angles, and always check your calculator is in degree mode.
• Verify answers with Pythagoras or \( \tan\theta = \sin\theta / \cos\theta \).

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