Right-Angled Trigonometry

While the hypotenuse is a fixed side, the "opposite" and "adjacent" labels will swap depending on which non-right angle you choose as your reference.

The SOH CAH TOA Ratios

SOH CAH TOA is a mnemonic used to remember the three primary trigonometric ratios. These formulas link the side lengths of a right-angled triangle to its interior angles:

$$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$ $$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$ $$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$

Calculating Missing Side Lengths

To calculate a missing side length, you must identify the correct trigonometric ratio, substitute the known values, and use algebraic rearrangement to solve for the unknown side.

Calculate the length of the opposite side $x$ in a right-angled triangle where the reference angle is $50^\circ$ and the adjacent side is $6\text{ cm}$.

Step 1: Label the sides relative to the angle.

• Known side ($6\text{ cm}$) = Adjacent ($A$)
• Unknown side ($x$) = Opposite ($O$)

Step 2: Select the correct ratio using SOH CAH TOA.

• We have $O$ and $A$, so we use TOA: $\tan(\theta) = \frac{O}{A}$.

Step 3: Substitute the values into the formula.

• $\tan(50^\circ) = \frac{x}{6}$

Step 4: Rearrange the formula to make $x$ the subject.

• $x = 6 \times \tan(50^\circ)$

Step 5: Calculate the final answer.

• $x \approx 7.15\text{ cm}$ (to 3 sig figs).

When the unknown side is the denominator in your fraction, the algebraic rearrangement requires an extra step.

Calculate the length of the hypotenuse $x$ in a right-angled triangle where the angle is $34^\circ$ and the opposite side is $4\text{ cm}$.

Step 1: Label the sides relative to the angle.

• Known side ($4\text{ cm}$) = Opposite ($O$)
• Unknown side ($x$) = Hypotenuse ($H$)

Step 2: Select the correct ratio using SOH CAH TOA.

• We have $O$ and $H$, so we use SOH: $\sin(\theta) = \frac{O}{H}$.

Step 3: Substitute the values into the formula.

• $\sin(34^\circ) = \frac{4}{x}$

Step 4: Rearrange algebraically by "swapping" the unknown variable and the trigonometric function.

• $x = \frac{4}{\sin(34^\circ)}$

Step 5: Calculate the final answer.

• $x \approx 7.15\text{ cm}$ (to 3 sig figs).

Calculating Missing Angles

To find a missing angle rather than a side, you must use an inverse trigonometric function ($\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$). While standard trigonometric ratios output side proportions, inverse functions return the principal value of the missing angle in degrees.

Calculate the missing angle $x$ in a right-angled triangle where the opposite side is $7\text{ cm}$ and the hypotenuse is $12\text{ cm}$.

Step 1: Label the sides relative to the angle.

• We have the Opposite ($7\text{ cm}$) and the Hypotenuse ($12\text{ cm}$).

Step 2: Select the correct ratio using SOH CAH TOA.

• Use SOH: $\sin(x) = \frac{O}{H}$.

Step 3: Substitute the known values.

• $\sin(x) = \frac{7}{12}$

Step 4: Apply the inverse sine function to find $x$.

• $x = \sin^{-1}(\frac{7}{12})$

Step 5: Calculate and round the final answer.

• $x = 35.6853...^\circ$
• $x = 35.7^\circ$ (to 1 d.p.).

Calculator Settings and Accuracy

For Edexcel GCSE Mathematics, your calculator must strictly be in Degree mode, typically indicated by a 'D' or 'DEG' at the top of the display.

• If your calculator is set to Radians (R) or Gradians (G), all trigonometric calculations will be completely incorrect.
• To maintain accuracy, do not write down truncated decimal values halfway through a calculation. Use the exact fraction or the 'ANS' button on your calculator until the final step.