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Wize High School Grade 10 Math Textbook > Trigonometry of Right Triangles

Right Triangle Word Problems

Solving Right TrianglesPrevious Section
Sine LawNext Section
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Did you know?

A sextant is a tool that helps you measure the angle between an object and the horizon.

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Example
Using a sextant, we can measure the angle between the top of this windmill and the horizon. Suppose you do not have a super tall ladder and measuring tape that allows you to get to the top of the windmill. How can you safely determine the height of the windmill?

You can measure the horizontal distance between the base of the windmill and the person using the sextant to measure the angle, then we can use the tan⁡\tan ratio to determine the height of the windmill.
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Right Angle Triangle Problems

Angle of Elevation & Depression

An angle of elevation is the angle from the horizontal upward to an object.

An angle of depression is the angle from the horizontal downward to an object.

3D Word Problem

Sometimes you will be given questions that involve a triangle in 3D, you can draw multiple triangles to help you determine the side length of a triangle in 3D.
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Example: Right Angle Triangle Problem

Bineshii is standing on a platform in front of a totem pole. The angle of elevation from Bineshii to the top of the totem pole is 50°50\degree, the angle of depression from Bineshii to the bottom of the totem pole is 20°20\degree. The horizontal distance between the base of the totel pole to the base of the platform that Bineshii is standing on is 4.1 meters. Find the height of the totem pole.
Adapted from | CC By 2.0

Finding x\bco x

Using the triangle with the 20°20\degreeangle, the opposite side is xx, the adjacent side is 4.1m4.1m, so we should use the tan⁡\tan ratio:
tan⁡20°=x4.1tan⁡20°×4.1=x0.364×4.1≈x1.49≈xx≈1.49m\begin{array}{rcl} \tan 20\degree&=&\dfrac{x}{4.1}\\\\ \tan20\degree\times4.1&=&x\\\\ 0.364\times4.1&\approx&x\\\\ 1.49&\approx&x\\\\ x&\approx&1.49m \end{array}

Finding y\bco y

Using the triangle with the 50°50\degree angle, the opposite side is yy, the adjacent side is 4.1m4.1m, so we should use the tan⁡\tan ratio:
tan⁡50°=y4.1tan⁡50°×4.1=y1.192×4.1≈y4.89≈yy≈4.89m\begin{array}{rcl} \tan 50\degree&=&\dfrac{y}{4.1}\\\\ \tan50\degree\times4.1&=&y\\\\ 1.192\times4.1&\approx&y\\\\ 4.89&\approx&y\\\\ y&\approx&4.89m \end{array}

Finding the height of the totem pole

Therefore, the height of the totem pole is 1.49m+4.89m≈6.38m1.49m+4.89m\approx\boxed{6.38m}

Practice: Solving Right Triangles

A triangular field is enclosed by fences as shown in the picture below. Find the area of the entire triangular field,

Practice: Solving Right Triangles

Determine the value of θ\theta in the picture below.

Practice: Solving Right Triangles

A cube has side lengths 1m1m, find the angle between the diagonal and the base of the cube

Practice: Solving Right Triangles

A 100m tall building is standing next to a taller building that is a certain distance apart. The angle of elevation from the top of the shorter building to the top of the taller building is 30°30\degree, and the angle of depression from the top of the shorter building to the bottom of the taller building is 40°40\degree .

a) Determine the distance between the two buildings

b) Determine the height of the taller building.