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Solving a right triangle involves calculating all angle measures and side lengths. To do this, the Pythagorean Theorem and trigonometric ratios can be used. The methods used to find all angle measures and side lengths will change depending on the given information.

Solve the right triangle completely.

Show Solution

To begin, notice that two angles are known — $28_{∘}$ and $90_{∘}.$ The third angle can be found using the Interior Angles Theorem. $180_{∘}−28_{∘}−90_{∘}=62_{∘}$ Thus, the third angle measures $62_{∘}.$

What remains is to find the side lengths $x$ and $y.$ Since only one side length is given, it is not yet possible to use the Pythagorean Theorem. Instead, a trigonometric ratio can be used. We can choose either angle to use. Let's use $28_{∘}.$ Relative to this angle, the adjacent side is known and $x$ is the opposite side. Thus, tangent can be used to determine $x.$$tan(θ)=adjopp $

SubstituteValuesSubstitute values

$tan(28_{∘})=1.2x $

UseCalcUse a calculator

$0.531…=1.2x $

MultEqn$LHS⋅1.2=RHS⋅1.2$

$0.638…=x$

RearrangeEqnRearrange equation

$x=0.638…$

$a_{2}+b_{2}=c_{2}$

SubstituteValuesSubstitute values

$1.2_{2}+0.64_{2}=c_{2}$

CalcPowCalculate power

$1.44+0.4096=y_{2}$

AddTermsAdd terms

$1.8496=y_{2}$

SqrtEqn$LHS =RHS $

$±1.36=y$

$y>0$

$1.36=y$

RearrangeEqnRearrange equation

$y=1.36$

Right triangles can be used to solve problems in context. Sometimes, it can be helpful to use angles of elevation or angles of depression.

Suppose someone is standing on the ground looking up toward the top of a tree. The angle made between the horizon and the viewer's gaze as they look up is an angle of elevation.

Here, the angle of elevation is marked $θ.$

Suppose someone is standing on a cliff looking down toward a ship in the ocean below. The angle made between the horizon and the viewer's gaze as they look down is an angle of depression.

Here, the angle of depression is labeled $θ.$

Martina is sitting on a bench in a harbor. A sail boat lies $40$ meters away from her. On top of the boat's $11$ meter high mast there is a seagull. Find the angle of elevation and depression when Martina and the seagull look at each other.

Show Solution

To begin, we can make a sketch to help us visualize the situation. Martina is labeled $M,$ the boat $B,$ and the seagull $S.$

The angle of elevation, $θ,$ is the angle looking up from the horizon. Since the seagull is higher than Martina, $θ$ will be the angle made when Martina looks up toward the seagull. Let's add this line to the diagram.

$θ$ can be calculated using trigonometric ratios. Since the opposite and adjacent sides, relative to $θ,$ are known, we can use $tan(θ).$ Since $tan(θ)=4011 ,$ it follows that $θ=tan_{-1}(4011 ).$ $θ=tan_{-1}(4011 )≈15_{∘}$ Thus, the angle of elevation is approximately $15_{∘}.$ The angle of depression, $α,$ is the angle looking down from the horizon. Since Martina is lower than the seagull, $α$ will be the angle made when the seagull looks down toward the Martina. Let's add this line to the diagram.Notice that the two horizont lines are parallel and that the line of sight is a transversal. By the Alternate Interior Angles Theorem $α$ and $θ$ are congruent. Therefore, the angle of depression is also approximately $15_{∘}.$

$Angle of elevationAngle of depression ≈15_{∘}.≈15_{∘} $

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