Sketching Diagrams

Lesson 3: Solving Problems using Trigonometry - Sketching Diagrams

If you need to solve a problem that requires trigonometry, it is important to sketch a diagram before trying to solve the problem. A diagram will help avoid the use of incorrect trigonometric ratios.

As with all problem solving you may be required to perform further calculations to ensure that you are meeting the requirements of the question. The steps for solving a trigonometry problem are:

Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated. Sketch a diagram if one is not provided.

Step 2: State the appropriate ratio.

Step 3: Substitute known values and calculate the unknown value.

Step 4: Complete the calculations required to answer the question asked.

   Multimedia

A video showing the solution to a trigonometric problem without a diagram is provided.

EXAMPLE 1

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Jon wants to walk from one corner of a square park, of length 100 m, to the opposite corner. How much farther, to the nearest tenth of a metre, is it to walk around the outside of the park than it is to cut directly across the park at a 45° angle?

Solution

Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated. Sketch a diagram if one is not provided.

Begin by sketching a diagram to represent the situation.



Step 2: State the appropriate ratio.

\(\text{sin}\,\theta=\frac{\text{length opposite}\,\theta}{\text{hypotenuse}}\)

Either the sine ratio or the cosine ratio because the lengths of both legs of the right triangle are known.

Step 3: Substitute known values and calculate the unknown value.

\(\begin{align} \text{sin}\,\theta&=\frac{\text{opp}}{\text{hyp}} \\ \\ \text{sin}\,45°&=\frac{100\,\text{m}}{d} \\ \\ d\times \text{sin}\,45°&=\frac{100\,\text{m}}{\cancel{d}}\times \cancel{d} \\ \\ d\times \text{sin}\,45°&=100\,\text{m} \\ \\ \frac{d\times \cancel{\text{sin}\,45°}}{\cancel{\text{sin}\,45°}}&=\frac{100\,\text{m}}{\text{sin}\,45°} \\ \\ d&=141.4\,\text{m} \\ \end{align}\)

Step 4: Complete the calculations required to answer the question asked.

Each side of the park is 100m in length, so if Jon walked around the outside of the park, he would have to walk 2 × 100m = 200m.

The diagonal distance straight through the park is approximately 141.4m.

200m - 141.4m = 58.6m

It is approximately 58.6m farther to walk around the park.

EXAMPLE 2

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Step 2: State the appropriate ratio.

The length adjacent to θ and the hypotenuse are known, so use the cosine ratio.

\(\text{cos}\,\theta=\frac{\text{length adjacent to}\,\theta}{\text{hypotenuse}}\)

Step 3: Substitute known values and calculate the unknown value.

\(\begin{align} \text{cos}\,\theta&=\frac{\text{length adjacent to}\,\theta}{\text{hypotenuse}} \\ \\ \text{cos}\,\theta&=\frac{2.9\,\text{m}}{7.4\,\text{m}} \\ \\ \theta&=\text{cos}^{-1}\left(\frac{2.9}{7.4}\right) \\ \\ \theta&=67° \\ \end{align}\)

The angle between the guy wire and the ground is approximately 67°.




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