Trigonometry

The One Thing

Trigonometry solves triangles by matching the information given to the appropriate relationship: Pythagoras’ theorem and SOHCAHTOA for right-angled triangles, and the sine rule, cosine rule or triangle-area formula for general triangles. Bearings extend these methods to accurately describe directions as three-figure angles measured clockwise from north.

Definitions and Results

• Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
• Pythagoras’ theorem: In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:

$$a^2+b^2=c^2$$ where $c$ is the hypotenuse. This requires a right-angled triangle.

• Opposite: The side directly opposite the angle being considered.
• Adjacent: The side next to the angle being considered, excluding the hypotenuse.
• Sine ratio:

$$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}$$ for an angle in a right-angled triangle.

• Cosine ratio:

$$\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}$$ for an angle in a right-angled triangle.

• Tangent ratio:

$$\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}$$ for an angle in a right-angled triangle.

• Inverse trigonometric functions: ${\sin }^{-1}$, ${\cos }^{-1}$ and ${\tan }^{-1}$, used to find an unknown angle from a ratio.
• Bearing: A three-figure angle measured clockwise from north, usually written with three digits, such as $065^{\circ }$. In particular, $090^{\circ }$ represents east, $180^{\circ }$ represents south and $270^{\circ }$ represents west.
• Sine rule: For any triangle,

$$\frac{a}{\sin (A)}=\frac{b}{\sin (B)}=\frac{c}{\sin (C)}$$ where each side is opposite its matching angle. It is especially useful when an opposite side–angle pair is known.

• Cosine rule: For any triangle,

$$a^2=b^2+c^2-2bc\cos (A)$$ where $a$ is opposite angle $A$. It is useful when two sides and the included angle are known, or when all three sides are known. To find an angle: $$\cos (A)=\frac{b^2+c^2-a^2}{2bc}.$$

• Area of a triangle: If two sides and their included angle are known,

$$\text{area}=\frac{1}{2}ab\sin (C),$$ where $C$ is the included angle between sides $a$ and $b$.

• Angle of elevation: The angle measured upward from a horizontal line to an object.
• Angle of depression: The angle measured downward from a horizontal line to an object.
• Back bearing: The bearing in the reverse direction. If the original bearing is less than $180^{\circ }$, add $180^{\circ }$; if it is $180^{\circ }$ or more, subtract $180^{\circ }$.

Worked Methods

1. Solving a right-angled triangle using Pythagoras’ theorem

• Confirm that the triangle is right-angled.
• Identify the hypotenuse, which is opposite the right angle.
• Use

$$a^2+b^2=c^2.$$

• To find a missing shorter side, rearrange:

$$a=\sqrt{c^2-b^2}.$$

• Substitute the known lengths and calculate.
• Give the answer with suitable units and accuracy.

2. Solving a right-angled triangle using SOHCAHTOA

• Confirm that the triangle is right-angled.
• Identify the given angle.
• Label the sides relative to that angle as opposite, adjacent and hypotenuse.
• Choose the ratio that contains the known and unknown sides:

$$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}},$$ $$\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}},$$ $$\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}.$$

• Substitute the known values and rearrange to find the missing length.
• For an unknown angle, use the appropriate inverse function:

$$\theta ={\sin }^{-1}(\text{ratio}),\theta ={\cos }^{-1}(\text{ratio}),\theta ={\tan }^{-1}(\text{ratio}).$$

• Ensure the calculator is in degree mode when angles are measured in degrees.

3. Using the sine rule

• Draw the triangle and label each side with the capital-letter angle opposite it:

$a$ opposite $A$, $b$ opposite $B$, and $c$ opposite $C$.

• Identify a known opposite side–angle pair.
• Use

$$\frac{a}{\sin (A)}=\frac{b}{\sin (B)}=\frac{c}{\sin (C)}.$$

• Select the two matching fractions containing the known values and the required side or angle.
• Rearrange and substitute carefully.
• Check that every side is matched with its opposite angle.
• Round only the final answer, retaining extra calculator digits during the calculation.

4. Using the cosine rule to find a side

• Identify the two known sides and their included angle.
• Label the side opposite the included angle as $a$, with the corresponding sides $b$ and $c$.
• Use

$$a^2=b^2+c^2-2bc\cos (A).$$

• Substitute the known side lengths and angle.
• Calculate $a^2$, then take the positive square root to find $a$.
• Give the answer with suitable units and accuracy.

5. Using the cosine rule to find an angle

• Identify the side $a$ opposite the required angle $A$, and the other two sides $b$ and $c$.
• Use the rearranged formula:

$$\cos (A)=\frac{b^2+c^2-a^2}{2bc}.$$

• Substitute the three side lengths.
• Apply the inverse cosine function:

$$A={\cos }^{-1}\left(\frac{b^2+c^2-a^2}{2bc}\right).$$

• Ensure the calculator is in degree mode and round the final angle appropriately.

6. Finding the area of a triangle

• Identify two known sides, $a$ and $b$.
• Identify their included angle, $C$.
• Use

$$\text{area}=\frac{1}{2}ab\sin (C).$$

• Substitute the side lengths and included angle.
• Calculate and give the answer in square units.

7. Working with bearings

• Draw a clear diagram showing north at the relevant points.
• Measure the direction clockwise from north.
• Write the bearing as a three-figure angle, including leading zeroes where necessary; for example, $065^{\circ }$.
• Use the reference directions $090^{\circ }$ for east, $180^{\circ }$ for south and $270^{\circ }$ for west.
• For a back bearing, add $180^{\circ }$ if the original bearing is less than $180^{\circ }$, or subtract $180^{\circ }$ if it is $180^{\circ }$ or more.
• If the bearing problem also involves lengths or angles in a triangle, choose Pythagoras, SOHCAHTOA, the sine rule or the cosine rule according to the information provided.

Where It Goes Wrong

• Applying Pythagoras’ theorem or SOHCAHTOA to a triangle that is not right-angled; these methods require a right-angled triangle.
• Misidentifying the hypotenuse, which must be the longest side and opposite the right angle.
• Confusing the opposite and adjacent sides, particularly because the labels depend on the angle being considered.
• Matching a side with the wrong angle when using the sine rule or cosine rule; each side must correspond to its opposite angle.
• Forgetting that a bearing is measured clockwise from north and must be written using three figures.
• Using radians instead of degrees, rounding intermediate values too early, or omitting suitable units and accuracy from the final answer.

What Gets Asked

• Find a missing side in a right-angled triangle using Pythagoras’ theorem.
• Find a missing side or angle in a right-angled triangle using SOHCAHTOA and inverse trigonometric functions.
• Use the sine rule to calculate a missing side or angle when an opposite side–angle pair is known.
• Use the cosine rule to calculate a missing side from two sides and the included angle, or a missing angle from three sides.
• Calculate the area of a triangle using $\frac{1}{2}ab\sin (C)$.
• Interpret or calculate angles of elevation and angles of depression.
• Calculate bearings, back bearings and directions using three-figure angles measured clockwise from north.
• Set out a complete solution by drawing a diagram, selecting the correct rule, substituting accurately, using degree mode, retaining calculator precision and giving an answer with suitable units and rounding.