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Using Tools and Trigonometry to Measure Heights

Ramsha is an architecture student in New York. She wants to measure the height of Flatiron Building using a tape measure, protractor, and laser pointer.

How could she use these tools to measure the height? In how many different ways can she measure the height? Does the distance from the building affect the measuring?

Example

Measuring the Angle of Elevation of a Flight of Stairs

LaShay is planning to build stairs for the second floor of her house. The horizontal part of a step is called the tread. The vertical part is called the riser. The tread and riser are fixed into a stringer.

She knows that the recommended riser-to-thread ratio is $7 inches : 11 inches .$ With this information, what should be the measure of the angle between the stairs and the ground? Round the answer to the nearest degree.

Hint

Since the tread is the horizontal part, it is parallel to the ground. By that same logic, the riser is perpendicular to the ground.

Solution

Recall that tread is the horizontal part and riser is the vertical part of a step. Therefore, tread is parallel to the ground while the riser is perpendicular.

Using this information and the Consecutive Interior Angles Theorem, it can be concluded that the angle between the tread and riser is a right angle.

Consequently, the tread, riser, and stringer form a right triangle. Additionally, by the Alternate Interior Angles Theorem, the top-left acute angle of the triangle is also $x^{\circ} .$

Because the ratio of the riser to the tread is $7 : 11,$ the tangent ratio can be used to write an equation for $x .$

tan x^{\circ} = \frac{riser}{tread} \Rightarrow tan x^{\circ} = \frac{7}{1 1}

Finally, by using the inverse of the tangent ratio, the value of $x$ can be found.

tan x^{\circ} = \frac{7}{1 1} \Leftrightarrow x^{\circ} = tan^{- 1} (\frac{7}{1 1}) \approx 3 2^{\circ}

Therefore, the measure of the angle between the stairs and ground is about

3 2^{\circ} .

Closure

Using Trigonometry in Astronomy

In this lesson, a few real-life applications of trigonometric ratios have been detailed. There are many more fields from video games to astronomy where trigonometric ratios play essential roles. Take for example, our Solar System and galaxy, astronomers use trigonometry to find the distance to stars and planets!

Astronomers measure the location of a star in the sky at one point of the year. They then measure again six months later when the Earth is on the opposite side of the Sun. In the meantime, the star moves a tiny amount compared to Earth — a phenomenon known as parallax.

Because the distance from one side of the orbit of Earth to the other is known, the angles can be calculated. Even cooler, the distance to the star can then be computed using trigonometric ratios. Before completing this lesson, think about how trigonometric ratios can be used in other areas.

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Hint

Solution