Some Applications of Trigonometry

The One Thing

Heights and distances are found by modelling the situation as a right triangle, identifying the angle of elevation or depression, and applying the trigonometric ratio that connects the known and unknown sides. A clear diagram, correct angle interpretation, consistent units, and appropriate treatment of observer’s eye level are essential.

Definitions and Results

• Line of sight: The straight line from the observer’s eye to the point being viewed.

• Angle of elevation: The angle formed between the horizontal line through the observer’s eye and the line of sight to an object above the observer. It is measured upward from the horizontal.

• Angle of depression: The angle formed between the horizontal line through the observer’s eye and the line of sight to an object below the observer. It is measured downward from the horizontal.

• Height: The vertical distance from the base of an object to its top or another specified point.

• Distance: Usually, the horizontal distance between the observer and the foot of the object.

• Angle of elevation and angle of depression relationship: When horizontal lines are parallel, the angle of depression from one point equals the corresponding angle of elevation from the other point. This permits an angle of depression to be used as the corresponding angle in the right triangle.

• Tangent ratio: For an angle in a right triangle,

$$\tan \theta =\frac{\text{opposite side}}{\text{adjacent side}}.$$ It is commonly used when height and horizontal distance are involved.

• Sine ratio: For an angle in a right triangle,

$$\sin \theta =\frac{\text{opposite side}}{\text{hypotenuse}}.$$ It is used when the opposite side and hypotenuse are related.

• Cosine ratio: For an angle in a right triangle,

$$\cos \theta =\frac{\text{adjacent side}}{\text{hypotenuse}}.$$ It is used when the adjacent side and hypotenuse are related.

• Observer’s eye level: The height of the observer’s eyes above the ground. It must be included when the line of sight begins above ground level.

• Right-triangle side identification: The hypotenuse is opposite the right angle; the perpendicular is opposite the chosen angle; and the base is adjacent to the chosen angle.

• Primary trigonometric ratios: For an angle $\theta$ in a right triangle,

$$\sin \theta =\frac{\text{perpendicular}}{\text{hypotenuse}},\cos \theta =\frac{\text{base}}{\text{hypotenuse}},\tan \theta =\frac{\text{perpendicular}}{\text{base}}.$$

• Reciprocal ratios:

$$\mathrm{cosec}\theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{1}{\tan \theta }.$$

• Pythagorean identity:

$${\sin }^2\theta +{\cos }^2\theta =1.$$ This may be used to check or find trigonometric values.

• Tangent–secant identity:

$$1+{\tan }^2\theta ={\sec }^2\theta .$$ This may be used when tangent and secant are involved.

• Height and distance formulae: If an object has height $h$, horizontal distance $d$, and angle of elevation $\theta$, then

$$\tan \theta =\frac{h}{d},h=d\tan \theta ,d=\frac{h}{\tan \theta }.$$

• Total height with observer’s eye level: If the observer’s eye is at height $e$, the angle of elevation to the top is $\theta$, and the horizontal distance is $d$, then

$$\text{total height}=e+d\tan \theta .$$

• Standard trigonometric values:

$$\sin 30^{\circ }=\frac{1}{2},\cos 30^{\circ }=\frac{\sqrt{3}}{2},\tan 30^{\circ }=\frac{1}{\sqrt{3}};$$ $$\sin 45^{\circ }=\cos 45^{\circ }=\frac{1}{\sqrt{2}},\tan 45^{\circ }=1;$$ $$\sin 60^{\circ }=\frac{\sqrt{3}}{2},\cos 60^{\circ }=\frac{1}{2},\tan 60^{\circ }=\sqrt{3}.$$

Worked Methods

Method 1: Modelling a height or distance with tangent

• Draw a labelled diagram of the situation.
• Represent the object’s height as a vertical side and the horizontal distance from the observer to the foot of the object as a horizontal side.
• Mark the right angle, line of sight, angle of elevation or depression, height, and distance.
• Identify the perpendicular and base relative to the given angle.
• Use

$$\tan \theta =\frac{\text{perpendicular}}{\text{base}}.$$

• If the object has height $h$ and horizontal distance $d$, write

$$\tan \theta =\frac{h}{d}.$$

• Rearrange as required:

$$h=d\tan \theta$$ when finding height, or $$d=\frac{h}{\tan \theta }$$ when finding distance.

• Give the answer in appropriate units and check whether it is reasonable.

Method 2: Including the observer’s eye level

• Draw the vertical object and the horizontal distance from the observer to its base.
• Mark the observer’s eye level $e$.
• Treat the vertical distance from the observer’s eye to the top of the object as the opposite side of the right triangle.
• Use the angle of elevation $\theta$ and horizontal distance $d$:

$$\tan \theta =\frac{\text{height above eye level}}{d}.$$

• Find the height above eye level:

$$\text{height above eye level}=d\tan \theta .$$

• Add the observer’s eye level:

$$\text{total height}=e+d\tan \theta .$$

Method 3: Using sine

• Draw and label the right triangle.
• Identify the chosen angle, the opposite side, and the hypotenuse.
• Use

$$\sin \theta =\frac{\text{opposite side}}{\text{hypotenuse}}.$$

• Rearrange the equation to find the unknown side.
• Use sine when the opposite side and hypotenuse are the relevant known and unknown quantities.

Method 4: Using cosine

• Draw and label the right triangle.
• Identify the chosen angle, the adjacent side, and the hypotenuse.
• Use

$$\cos \theta =\frac{\text{adjacent side}}{\text{hypotenuse}}.$$

• Rearrange the equation to find the unknown side.
• Use cosine when the horizontal distance or another adjacent side is related to the line of sight, which is the hypotenuse.

Method 5: Solving an angle of depression problem

• Draw the observer’s horizontal line and the horizontal line through the lower point.
• Mark the line of sight from the observer to the lower point.
• Identify the angle of depression as the angle measured downward from the observer’s horizontal.
• Use the fact that horizontal lines are parallel: the angle of depression equals the corresponding angle of elevation at the lower point.
• Use this corresponding angle in the right triangle.
• Select the appropriate ratio, commonly

$$\tan \theta =\frac{\text{height}}{\text{horizontal distance}}.$$

• Solve and attach the correct units.

Method 6: Selecting values and using a calculator

• Use the given angle in degrees.
• Select the trigonometric ratio that connects the known and unknown sides.
• For standard angles, use the stated values for $30^{\circ }$, $45^{\circ }$, and $60^{\circ }$.
• For non-standard angles, use a calculator.
• Ensure that the calculator is set to degree mode.
• Substitute accurately and round only as appropriate for the question.

Method 7: Checking a solution

• Confirm that the diagram contains the right angle, line of sight, angle, height, distance, and eye level where relevant.
• Check that the selected ratio connects the correct sides.
• Check that all measurements use consistent units, such as metres for both height and distance.
• Verify that an angle of elevation has been measured upward and an angle of depression downward.
• Use

$${\sin }^2\theta +{\cos }^2\theta =1$$ or $$1+{\tan }^2\theta ={\sec }^2\theta$$ where appropriate.

• Check reasonableness: for the same horizontal distance, a taller object or a larger angle generally corresponds to a greater height.

Where It Goes Wrong

• The diagram is omitted or inadequately labelled, so the right angle, line of sight, height, distance, angle, or observer’s eye level is misidentified.
• The angle of elevation is confused with the angle of depression; elevation is measured upward from the horizontal, whereas depression is measured downward.
• The wrong trigonometric ratio is selected instead of matching the known and unknown sides: tangent connects height and horizontal distance, sine connects the opposite side and hypotenuse, and cosine connects the adjacent side and hypotenuse.
• The angle of depression is used without applying the parallel-horizontal-lines relationship that gives the corresponding angle of elevation.
• The observer’s eye level is forgotten when the line of sight begins above ground level; the total height is $e+d\tan \theta$, not merely $d\tan \theta$.
• Units are mixed, or a calculator is used in radian mode instead of degree mode.

What Gets Asked

This material supports questions requiring students to:

• Find the height of an object from a horizontal distance and an angle of elevation using

$$h=d\tan \theta .$$

• Find the horizontal distance from a known height and angle using

$$d=\frac{h}{\tan \theta }.$$

• Find an object’s total height when the observer’s eye level is given using

$$e+d\tan \theta .$$

• Solve problems involving an angle of depression by converting it to the corresponding angle of elevation.
• Decide whether sine, cosine, or tangent is the appropriate ratio from the sides known and required.
• Use the line of sight and a horizontal or vertical reference to convert a real-life situation into a right-triangle problem.
• Evaluate expressions using the standard values for $30^{\circ }$, $45^{\circ }$, and $60^{\circ }$, or use a calculator for non-standard angles.
• Apply or verify

$${\sin }^2\theta +{\cos }^2\theta =1$$ and $$1+{\tan }^2\theta ={\sec }^2\theta .$$

• Interpret and use the reciprocal ratios $\mathrm{cosec}\theta$, $\sec \theta$, and $\cot \theta$.
• Present a complete solution with a labelled diagram, accurate substitution, consistent units, and a reasonableness check.