Solving Right Triangles and Applied Problems

Core Concepts

Calculate missing sides and angles, angles of elevation, and angles of depression.

Solving Right Triangles and Applied Problems

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What does it mean to 'solve a right triangle'?

To determine the lengths of all three sides and the measures of all three angles given at least one side and one other part (either a side or an acute angle).

In a right triangle with acute angle $35^{\circ}$ and adjacent side $14$, what equation solves for the opposite side $x$?

$$\tan 35^{\circ} = \frac{x}{14} \implies x = 14\tan 35^{\circ}$$

In a right triangle with acute angle $22^{\circ}$ and opposite side $8$, what equation solves for the hypotenuse $h$?

$$\sin 22^{\circ} = \frac{8}{h} \implies h = \frac{8}{\sin 22^{\circ}}$$

How is an unknown acute angle $\theta$ solved when the opposite side is $5$ and the hypotenuse is $11$?

$$\sin\theta = \frac{5}{11} \implies \theta = \arcsin\left(\frac{5}{11}\right) = \sin^{-1}\left(\frac{5}{11}\right)$$

In $\triangle ABC$ with $\angle C = 90^{\circ}$, leg $a = 5$ and leg $b = 12$. Find angle $A$ symbolically and the hypotenuse $c$.

$$c = \sqrt{5^2 + 12^2} = 13$$ $$A = \arctan\left(\frac{5}{12}\right)$$

A $20\text{-ft}$ ladder leans against a vertical wall, making a $65^{\circ}$ angle with the ground. How high up the wall does it reach?

Let $h$ be the height: $$\sin 65^{\circ} = \frac{h}{20} \implies h = 20\sin 65^{\circ} \approx 18.13\text{ ft}$$

A person stands $50\text{ m}$ from the base of a building. The angle of elevation to the top is $38^{\circ}$. Disregarding eye height, how tall is the building?

$$\tan 38^{\circ} = \frac{\text{height}}{50} \implies \text{height} = 50\tan 38^{\circ} \approx 39.06\text{ m}$$

A lighthouse keeper at a height of $120\text{ ft}$ spots a boat with an angle of depression of $15^{\circ}$. How far is the boat from the base of the lighthouse?

$$\tan 15^{\circ} = \frac{120}{d} \implies d = \frac{120}{\tan 15^{\circ}} \approx 447.85\text{ ft}$$

Why should you avoid using a calculated rounded value to find a second unknown in multi-step triangle solving?

Using rounded intermediate results introduces rounding error propagation. Always prefer using the original given values with the appropriate trigonometric ratio when possible.

A ramp rises $3\text{ m}$ over a horizontal run of $20\text{ m}$. What is the ramp's angle of incline $\theta$?

$$\tan\theta = \frac{3}{20} \implies \theta = \arctan\left(\frac{3}{20}\right) \approx 8.53^{\circ}$$

How do you find the area of a right triangle given hypotenuse $c$ and acute angle $\theta$?

$$\text{Leg}_1 = c\sin\theta, \quad \text{Leg}_2 = c\cos\theta$$ $$\text{Area} = \frac{1}{2}(c\sin\theta)(c\cos\theta) = \frac{1}{2}c^2\sin\theta\cos\theta$$

What is the angle of elevation?

Diagram illustrating the angle of elevation, formed between an observer's horizontal dashed line of sight and an upward line of sight directed toward a tree on a hill.

The angle formed by the horizontal line of sight and the observer's line of sight upward toward an object.

What is the angle of depression?

Diagram illustrating the angle of depression, showing an observer looking down from a cliff toward a tree, with the angle measured downward between the horizontal line of sight and the downward line of sight.

The angle formed by the horizontal line of sight and the observer's line of sight downward toward an object.

Why is an observer's angle of depression to an object on the ground equal to the object's angle of elevation back up to the observer?

Two parallel horizontal dashed lines intersected by a diagonal transversal line, highlighting equal alternate interior angles that represent the angle of depression and the angle of elevation.

Because the horizontal line of sight from the observer and the horizontal ground level are parallel, making the angle of depression and angle of elevation alternate interior angles.

Solving Right Triangles and Applied Problems

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