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4. Right Triangles & Trigonometry
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United States of AmericaTennessee
Integrated Math 3

4.06 Modeling with right triangles

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Lesson

Concept summary

The trigonometric ratios give us relationships between the sides and angles in a right triangle. In real life situations we won't always know all the information about a right triangle and we can use the trigonometric ratios, and their inverses, to find those missing values.

Real life applications often involve objects or points of reference that are at different heights. The following terms are used to define the angles between objects at different heights.

Angle of elevation

The angle between the horizontal and an object above it

A horizontal line has a line segment joining a point on the line to a point above the line. The acute angle between the line segment and the horizontal line is marked as the angle of elevation.
Angle of depression

The angle between the horizontal and an object below it

A horizontal line has a line segment joining a point on the line to a point below the line. The acute angle between the line segment and the horizontal line is marked as the angle of depression.

Worked examples

Example 1

A diagram of a person standing in an elevated stadium, and a singer standing in front of the person, at a lower position. A diagonal dashed segment is drawn from the person to the singer. A right triangle is drawn with the diagonal dashed segment as the hypotenuse, a vertical segment above the singer with length 20 meters, and a horizontal segment from the person to the point directly above the singer with length 68 meters. The angle formed by the hypotenuse, and the horizontal segment measures x.

Maria and her friends are at a rock concert, enjoying one of their favorite bands. The lead singer is 68 \text{ m} across and 20 \text{ m} down from where Maria is in the stadium:

Calculate the angle of depression, x, from Maria to the lead singer. Round your answer to two decimal places.

Approach

We want to look at the diagram and determine how the given information is related.

With respect to x, the angle of depression, we are given the opposite side and adjacent side of the right triangle. The opposite side has a length of 20 \text{ m} and the adjacent side has a length of 68 \text{ m}. Use this information to set up the appropriate triogonometric ratio and use the correct inverse trigonometric ratio to solve for x.

Solution

\displaystyle \tan x\displaystyle =\displaystyle \dfrac{20}{68}
\displaystyle x\displaystyle =\displaystyle \tan^{-1} \dfrac{20}{268}Take the inverse tangent of \dfrac{20}{68} to undo tangent
\displaystyle x\displaystyle =\displaystyle 16.39 \degreeEvaluate

The angle of depression from Maria to the lead singer is x=16.39 \degree

Reflection

We can see that the angle of elevation from the lead singer to Maria would also have a measure of 16 \degree as the two angles formed are alternate interior angles on parallel lines.

Example 2

Alberto is watching a cuckoo fly past. The angle of elevation from where he is standing to the bird is 28 \degree when it flies over a point which he measures to be 11 \text{ m} away from him.

Find the distance, between the Alberto and the cuckoo. Round your answer to two decimal places.

Approach

Right triangle A B C with right angle B. Side A B has a length of 11 meters. Angle A has a measure of 28 degrees.

We first want to draw a diagram to represent the given information. We know the horizontal distance from Alberto to the bird is 11 \text{ m} and the angle of elevation from Alberto to the cuckoo is 28 \degree.

We can use this information to write the appropriate trigonometric ratio and then solve for AC.

Solution

\displaystyle \cos\theta\displaystyle =\displaystyle \dfrac{\text{adjacent}}{\text{hypotenuse}}
\displaystyle \cos(28\degree)\displaystyle =\displaystyle \frac{11}{AC}Substitution
\displaystyle AC\cos(28\degree)\displaystyle =\displaystyle 11Multiply both sides by AC
\displaystyle AC\displaystyle =\displaystyle \frac{11}{\cos\left(28\degree\right)}Divide both sides by \cos\left(28\degree\right)
\displaystyle AC\displaystyle =\displaystyle 12.46Evaluate

The distance from Alberto to the cuckoo is 12.46 \text{ m}

Outcomes

M3.N.Q.A.1.B

Use appropriate quantities in formulas, converting units as necessary.

M3.G.SRT.C.5

Solve triangles.*

M3.G.SRT.C.5.A

Know and use the Pythagorean Theorem and trigonometric ratios (sine, cosine, tangent, and their inverses) to solve right triangles in a real-world context.

M3.G.SRT.C.5.B

Know and use relationships within special right triangles to solve problems in a real-world context.

M3.MP1

Make sense of problems and persevere in solving them.

M3.MP2

Reason abstractly and quantitatively.

M3.MP3

Construct viable arguments and critique the reasoning of others.

M3.MP4

Model with mathematics.

M3.MP5

Use appropriate tools strategically.

M3.MP6

Attend to precision.

M3.MP7

Look for and make use of structure.

M3.MP8

Look for and express regularity in repeated reasoning.

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