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Trigonometric ratios are often used to find the height or length of an object, or the distances between objects, indirectly. Therefore, they are applicable for various situations. In this lesson, the relation between the geometric shapes and trigonometric ratios will be investigated.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

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.

Flatiron Building

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

Placing a Ladder at the Correct Angle of Elevation

Heichi is painting his house. He has a $10 -$ feet long ladder. To reach the top of the wall, he needs to lean the ladder against the wall with a $6 0^{\circ}$ angle.

A 10-foot ladder rests its top against a wall at a 60-degree angle.

Find the distance from the bottom of the ladder to the wall.

Hint

Consider the trigonometric ratios.

Solution

Notice that the ladder, wall, and ground form a right triangle.

Since the distance is adjacent to the angle, the cosine ratio can be used to find the distance.

cos 6 0^{\circ} = \frac{x}{1 0}

Recall that the value of

cos 6 0^{\circ}

is

\frac{1}{2} .

Keeping that in mind, the value of

x

can be calculated.

cos 6 0^{\circ} = \frac{x}{1 0}

$cos 6 0^{\circ} = \frac{1}{2}$

\frac{1}{2} = \frac{x}{1 0}

$LHS \cdot 10 = RHS \cdot 10$

\frac{1 0}{2} = x

Calculate quotient

5 = x

Rearrange equation

x = 5

Therefore, the distance from the bottom of the ladder to the wall is

5

feet.

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} .

Example

Planning Constructions Using Trigonometry

Vincenzo plans to build a grain bin with a radius of $20$ feet. The slant of the roof will be $3 5^{\circ} .$ He wants the roof to overhang the edge of the bin by $2$ feet.

A grain bin has a cylindrical base with a 20 ft radius. Its top is a cone with a slant height of x, a base radius 2 ft larger than the cylinder, and the angle between its base radius and slant height is 35 degrees.

What should the slant height of the roof be? If necessary, round the answer to one decimal place.

Hint

What is the position of the slant height in relation to given angle? What information is needed to find the value of $x ?$

Solution

The grain bin consists of a cylinder and cone. The radius, altitude, and lateral face of the cone form a right triangle where the slant height is the hypotenuse.

Either the radius or the height of the cone is necessary to find the length of the slant height. By using the given information, the radius of the cone can be found immediately. Notice that the radius $r$ of the cone is the sum of the radius of the cylinder and the length of the roof overhang.

20 + 2 = 22 ft

Since the radius is the adjacent side of the given angle, cosine ratio can be used to find the value of $x .$

cos 3 5^{\circ} = \frac{2 2}{x}

$LHS \cdot x = RHS \cdot x$

x \cdot cos 3 5^{\circ} = 22

$LHS / cos 3 5^{\circ} = RHS / cos 3 5^{\circ}$

x = \frac{2 2}{cos 3 5 ^{\circ}}

Use a calculator

x = 26.857040 \dots

Round to $1$ decimal place(s)

x \approx 26.9

The slant height of the roof is about

26.9

feet.

Example

Using Trigonometry to Investigate the Real World

Maya is an archaeologist working in Egypt. Recently, she discovered the ruins of a pyramid. This is kind of a big deal. Even if most of the pyramid has eroded, Maya was able to determine that the length of a side of the square base is $94$ meters.

Pyramid

It is known that the ancient Egyptians built only two pyramids with faces inclined at $5 2^{\circ}$ angles. Maya thinks that she has just found one of them. Using the given information, find the height of the pyramid and round the answer to the nearest meter.

Hint

Complete the ruins of the pyramid. What is the position of the height of the pyramid in relation to the given angle?

Solution

By modeling the pyramid, the given angle and its height can be shown.

Notice that the altitude and an inclined face of the pyramid form a right angle with the base of the pyramid. Because the height is opposite the given angle, either the hypotenuse or the adjacent side is needed to find the height.

Since the length of a side of the square base is known, the adjacent side can be found by dividing it by $2 .$

\frac{9 4}{2} = 47

Now that the adjacent side has been found, the tangent ratio can be used to find the opposite side's value.

tan θ = \frac{opposite side}{adjacent side}

Substituting the values into this ratio, the height of the pyramid can be found.

tan θ = \frac{opposite side}{adjacent side}

SubstituteValues

Substitute values

tan 5 2^{\circ} = \frac{h}{4 7}

▼

Solve for

h

$LHS \cdot 47 = RHS \cdot 47$

47 \cdot tan 5 2^{\circ} = h

Rearrange equation

h = 47 \cdot tan 5 2^{\circ}

Use a calculator

h = 60.157256 \dots

Round to nearest integer

h \approx 60

The height of the pyramid that Maya discovered is about

60

meters.

Example

Industrial Applications of Trigonometry

A ball bearing consists of two concentric metal circles, called bearing races, separated by metal balls. The purpose of a ball bearing is to facilitate the movement between two interacting objects within a machines such as bicycles, wheelchairs, and even household items like washing machines.

A ball bearing is made up of two metal rings that are concentrically arranged and separated by small metal balls.

In the diagram, assume that there is no space between the metal balls. If the outer radius of the inner circle is $4$ inches, what would be the radius of a metal ball? If necessary, round the answer to one decimal place.

Hint

Draw a triangle with vertices at the center of the inner circle and at two adjacent outer metal balls. Then use the trigonometric ratios.

Solution

Begin by drawing a triangle whose vertices are the center of the inner circle and two adjacent metal balls.

Assume that the radius of each metal ball is $r$ inches.

Notice that

△ A B C

is an isosceles triangle where

A B = A C .

From here, to find the value of

r,

the measure of the vertex angle of

△ A B C

needs to be found. Since there are

12

metal balls, the measure of

∠ B A C

can be found by dividing

36 0^{\circ}

by

12 .

\frac{3 6 0 ^{\circ}}{1 2} = 3 0^{\circ}

Now that the measure of the vertex angle has been found, the altitude of the base can be drawn to aid the use of trigonometric ratios. Remember, the altitude of the base of an isosceles triangle bisects the vertex angle and the base therefore

3 0^{\circ}

can be split in half.

Finally, using the sine ratio in

△ A B D,

the value of

r

can be found.

sin θ = \frac{B D}{A B}

SubstituteValues

Substitute values

sin 1 5^{\circ} = \frac{r}{4 + r}

▼

Solve for

r

$LHS \cdot (4 + r) = RHS \cdot (4 + r)$

4 sin 1 5^{\circ} + r sin 1 5^{\circ} = r

$LHS - r sin 1 5^{\circ} = RHS - r sin 1 5^{\circ}$

4 sin 1 5^{\circ} = r - r sin 1 5^{\circ}

Factor out $r$

4 sin 1 5^{\circ} = r (1 - sin 1 5^{\circ})

$LHS / (1 - sin 1 5^{\circ}) = RHS / (1 - sin 1 5^{\circ})$

\frac{4 sin 1 5 ^{\circ}}{1 - sin 1 5 ^{\circ}} = r

Rearrange equation

r = \frac{4 sin 1 5 ^{\circ}}{1 - sin 1 5 ^{\circ}}

Use a calculator

r = 1.396792 \dots

Round to $1$ decimal place(s)

r \approx 1.4

Therefore, radius of each metal ball is about

1.4

inches.

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!

Solar System

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.

Solar System

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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