Applications of Right Triangles in Trigonometry

Applications of Right Triangles

Significant Digits in Trigonometric Calculations

Significant digits represent the actual precision of a measurement. In trigonometry, most function values and measurements are approximations, so it is important to consider the number of significant digits when reporting results.

Number of Significant Digits	Angle Measure to Nearest
2	Degree
3	Ten minutes, or nearest tenth of a degree
4	Minutes, or nearest hundredth of a degree
5	Tenth of a minute, or nearest thousandth of a degree

Solving Right Triangles

Given an Angle and a Side

To solve a right triangle when given one angle (other than the right angle) and one side, use trigonometric ratios to find the unknown sides and angles.

• Key Point: The sum of the angles in a triangle is 180°, so the other acute angle is found by subtracting the given angle from 90°.

• Formulas:

  • $\sin A = \frac{a}{c}$

  • $\cos A = \frac{b}{c}$

• Example: Given $A = 34^0'$ and $c = 12.7$ inches, find $a$ and $b$:

  • $\sin 34^0' = \frac{a}{12.7} \implies a = 12.7 \sin 34^0' \approx 7.19$ inches

  • $\cos 34^0' = \frac{b}{12.7} \implies b = 12.7 \cos 34^0' \approx 10.5$ inches

Given Two Sides

When two sides are known, use the Pythagorean theorem and inverse trigonometric functions to find the unknowns.

• Key Point: The Pythagorean theorem relates the sides: $b^2 = c^2 - a^2$

• Formulas:

  • $\sin A = \frac{a}{c}$

  • $A = \sin^{-1}\left(\frac{a}{c}\right)$

  • $B = 90^0' - A$

• Example: Given $a = 29.43$ cm, $c = 53.58$ cm:

  • $\sin A = \frac{29.43}{53.58} \implies A \approx 33.32^0$

  • $B = 90^0 - 33.32^0 = 56.68^0$

  • $b^2 = 53.58^2 - 29.43^2 \implies b \approx 44.77$ cm

Angles of Elevation and Depression

Definitions

• Angle of Elevation: The angle formed by a horizontal line and the line of sight to an object above the horizontal.

• Angle of Depression: The angle formed by a horizontal line and the line of sight to an object below the horizontal.

Solving Applied Trigonometry Problems

1. Draw a sketch and label it with the given information. Assign a variable to the unknown quantity.

2. Write an equation relating the given quantities to the variable using trigonometric relationships.

3. Solve the equation and check that your answer makes sense in the context of the problem.

Example: Finding the Angle of Elevation

• Problem: A building is 34.09 meters tall and casts a shadow 37.62 meters long. Find the angle of elevation of the sun.

• Solution:

  • $\tan B = \frac{34.09}{37.62}$

  • $B = \tan^{-1}\left(\frac{34.09}{37.62}\right) \approx 42.18^0$

Bearings

Methods for Expressing Bearings

• First Method: A single angle measured clockwise from north.

• Second Method: Uses a north-south reference line and an acute angle east or west from this line (e.g., N 42° E).

Example: Solving a Problem Involving Bearing (First Method)

• Given: Radar stations A and B are 3.70 km apart on an east-west line. Bearings to a plane from A and B are 61.0° and 331.0°, respectively.

• Find the distance from A to C using the cosine function:

  • $\cos 29.0^0 = \frac{b}{3.70}$

  • $b = 3.70 \cos 29.0^0 \approx 3.24$ km

Example: Solving a Problem Involving Bearing (Second Method)

• Given: Bearings from A to C is S 52° E, from A to B is N 84° E, from B to C is S 38° W. A plane flies at 250 mph for 2.4 hours from A to B (distance = 600 miles).

• Find the distance from A to C using the sine function:

  • $\sin 46^0 = \frac{b}{600}$

  • $b = 600 \sin 46^0 \approx 430$ miles

Applications: Airport Runway Bearings

• Runway numbers differ by 18 (180°) for left (L) and right (R) directions, numbered from 01 to 36.

• Example: If runway is labeled 12L, the opposite direction is 12 + 18 = 30, so the sign must read 30R.

• If the pilot turns right onto 30R, the bearing is 300°, indicating a northwest direction.

Calculating the Distance to a Star (Parallax Method)

Parallax is the apparent shift in position of a nearby object against a distant background when viewed from two different positions. In astronomy, it is used to calculate the distance to stars.

• Formula: $\sin \theta = \frac{d}{D}$, where $\theta$ is the parallax angle, $d$ is the Earth-Sun distance, and $D$ is the distance to the star.

• Example: For Alpha Centauri, $\theta \approx 0.000212^0$, Earth-Sun distance = 93,000,000 miles:

  • $d = \frac{93,000,000}{\sin 0.000212^0} \approx 2.51 \times 10^{13}$ miles

  • In light-years: $\frac{2.51 \times 10^{13}}{5.9 \times 10^{12}} \approx 4.3$ light-years

Solving a Problem Involving Angle of Elevation

• Problem: From a point, the angle of elevation to a tree is 36.7°. After moving back 50 ft, the angle is 22.2°. Find the height of the tree.

• Solution:

  • Let $x$ be the initial distance and $h$ the height.

  • From the first position: $h = x \tan 36.7^0$

  • From the second position: $h = (x + 50) \tan 22.2^0$

  • Set equal and solve for $x$:

  • $x \tan 36.7^0 = (x + 50) \tan 22.2^0$

  • $x = \frac{50 \tan 22.2^0}{\tan 36.7^0 - \tan 22.2^0}$

  • Substitute $x$ back to find $h$:

  • $h = x \tan 36.7^0 \approx 45$ feet

Graphing Calculator Solution

• Plot the lines $y = \tan 22.2^0 x$ and $y = \tan 36.7^0 (x - 50)$ and find their intersection to confirm $h \approx 45$ feet.