Applications of Trigonometry

Use trigonometry for right triangles, height and distance problems, components, and periodic models.

Trigonometry is useful because it turns angle information into measurements. When a problem has a right triangle, a direction, a rotating object, a wave, or a repeating pattern, trig gives a language for connecting the pieces.

Learning objectives

After this lesson you should be able to:

  • choose sine, cosine, or tangent from a right-triangle diagram;
  • solve height and distance problems using angles of elevation or depression;
  • split a vector into horizontal and vertical components;
  • identify amplitude, period, midline, and shift in a sinusoidal model;
  • check whether a numerical answer is reasonable;
  • use Python and SymPy to compute or verify application results.

The modeling routine

A trigonometry application is not just a calculation. It is a translation from words to a diagram and then from a diagram to an equation.

  1. Draw or imagine the right triangle or cycle. Mark known lengths and angles.
  2. Choose the unknown. Name the height, distance, component, or time value you want.
  3. Select a trig relationship. Use SOH-CAH-TOA for right triangles or a sinusoidal model for periodic behavior.
  4. Solve the equation. Use algebra and a calculator or exact values.
  5. Check units and size. A tree height should be positive; a component should not be longer than the vector; a period should match the context.

Right-triangle ratios

For an acute angle in a right triangle:

sin(angle) = opposite / hypotenuse
cos(angle) = adjacent / hypotenuse
tan(angle) = opposite / adjacent

A memory phrase is SOH-CAH-TOA.

The most important decision is not the formula itself; it is correctly labeling which side is opposite and which side is adjacent relative to the chosen angle.

Angles of elevation and depression

An angle of elevation is measured upward from a horizontal line. An angle of depression is measured downward from a horizontal line. In many textbook problems, the horizontal distance is adjacent to the angle and the vertical height is opposite the angle.

That means tangent often appears:

tan(angle) = height / horizontal distance
height = horizontal distance * tan(angle)

Components and navigation

If a quantity of length r points at an angle theta from the positive horizontal direction, its components are:

horizontal component = r cos(theta)
vertical component   = r sin(theta)

This is the same unit-circle idea scaled by r.

Periodic models

Sine and cosine model smooth repeating behavior, such as tides, daylight, temperature cycles, circular motion, and alternating current. A common model is:

y = A sin(B(t - C)) + D

where:

Symbol Meaning
|A| amplitude, or distance from midline to maximum
2pi / |B| period, or time for one full cycle
D midline, or average value
C horizontal shift

Worked example 1: height from angle of elevation

A student stands 40 meters from a building. The angle of elevation to the top is 35 degrees. Estimate the building height, ignoring the student’s eye height.

Step 1: Draw the triangle.

The horizontal distance is adjacent to the angle. The building height is opposite the angle.

Step 2: Choose a ratio.

Tangent uses opposite and adjacent:

tan(35 degrees) = height / 40

Step 3: Solve.

height = 40 tan(35 degrees)
height is about 28.0 meters

Step 4: Check reasonableness.

Since 35 degrees is less than 45 degrees, the height should be less than the horizontal distance 40 meters. The estimate 28.0 meters is reasonable.

Worked example 2: split a force into components

A 50 newton force is applied at 30 degrees above the horizontal. Find the horizontal and vertical components.

Step 1: Use the component formulas.

horizontal = 50 cos(30 degrees)
vertical = 50 sin(30 degrees)

Step 2: Use exact values.

cos(30 degrees) = sqrt(3)/2
sin(30 degrees) = 1/2

So:

horizontal = 25sqrt(3) newtons, about 43.3 newtons
vertical = 25 newtons

Step 3: Check.

Each component should be less than the original 50 newtons, and the horizontal component should be larger because the angle is closer to horizontal than vertical.

Worked example 3: build a sinusoidal model

A small tide varies between 2 meters and 8 meters. The time from one high tide to the next is 12 hours. Suppose high tide occurs at t=0.

Step 1: Find the midline.

midline = (maximum + minimum) / 2 = (8 + 2) / 2 = 5

Step 2: Find the amplitude.

amplitude = (maximum - minimum) / 2 = (8 - 2) / 2 = 3

Step 3: Find B from the period.

period = 2pi / |B| = 12
B = 2pi/12 = pi/6

Step 4: Choose cosine because the model starts at a maximum.

height = 3cos((pi/6)t) + 5

Check: at t=0, cos(0)=1, so the height is 8 meters, which matches high tide.

Common pitfalls

  • Labeling opposite and adjacent from the wrong angle.
  • Using degrees in a calculator or program that expects radians.
  • Forgetting to add eye height, sensor height, or starting height when the context includes it.
  • Reporting a component larger than the original vector.
  • Using a sine model when a cosine model would be simpler for a maximum or minimum starting point.
  • Forgetting units.

Practice exercises

  1. A ladder is 10 feet long and makes a 70 degree angle with the ground. How high up the wall does it reach?
  2. A ramp rises 2 meters over a horizontal distance of 12 meters. What is the angle of elevation?
  3. Find the components of a 20 newton force at 60 degrees above the horizontal.
  4. A Ferris wheel car is modeled by h(t)=15cos((pi/20)t)+18. What are the amplitude, midline, and period?
  5. A ship travels 100 km at 40 degrees north of east. Estimate the east and north components.
  1. Height is opposite the 70 degree angle and the ladder is the hypotenuse: height=10sin(70 degrees), about 9.4 feet.
  2. tan(theta)=2/12=1/6, so theta=arctan(1/6), about 9.5 degrees.
  3. Horizontal component: 20cos(60 degrees)=10 newtons; vertical component: 20sin(60 degrees)=10sqrt(3), about 17.3 newtons.
  4. Amplitude is 15, midline is h=18, and period is 2pi/(pi/20)=40 time units.
  5. East component: 100cos(40 degrees), about 76.6 km; north component: 100sin(40 degrees), about 64.3 km.

Subtopic guided practice and checkpoints

Lab 1: choose the right ratio

Guess first. If you know adjacent distance and want opposite height, which ratio should you use?

Guided exercise.

Use tangent because:

tan(angle) = opposite / adjacent

Checkpoint. If you know hypotenuse and want adjacent, which ratio should you use?

Lab 2: solve a height problem

Guess first. A pole casts a horizontal distance of 15 m from an angle of elevation of 30 degrees. Is the height greater or less than 15 m?

Guided exercise.

Because 30 degrees is less than 45 degrees, the height should be less than 15 m. Compute:

height = 15tan(30 degrees) = 15/sqrt(3), about 8.7 m

Checkpoint. Estimate the height for horizontal distance 20 m and angle 45 degrees.

Lab 3: compute components

Guess first. For a vector at 0 degrees, should the vertical component be positive, negative, or zero?

Guided exercise.

At 0 degrees, the vector points horizontally, so:

vertical = r sin(0 degrees) = 0
horizontal = r cos(0 degrees) = r

Checkpoint. For a 12 unit vector at 90 degrees, what are the horizontal and vertical components?

Lab 4: identify sinusoid parameters

Guess first. In y=4sin(2t)+7, what is the midline?

Guided exercise.

The +7 shifts the graph up, so the midline is:

y = 7

The amplitude is 4, and the period is:

2pi / 2 = pi

Checkpoint. For y=6cos((pi/3)t)-2, identify amplitude, midline, and period.

Lab 5: check units and reasonableness

Guess first. If a 30 m rope is used as a hypotenuse, can the vertical height be 35 m?

Guided exercise.

No. In a right triangle, the hypotenuse is the longest side. A height of 35 m would be impossible if the rope is only 30 m.

Checkpoint. If a 50 N force has components 60 N and 20 N, what warning sign do you notice?

  1. Cosine, because cos(angle)=adjacent/hypotenuse.
  2. height=20tan(45 degrees)=20 m.
  3. Horizontal component 0; vertical component 12.
  4. Amplitude 6, midline y=-2, period 2pi/(pi/3)=6.
  5. A component cannot be larger than the original vector, so the answer should be rechecked.

Applications guessing game

Using this lesson with edumath and SymPy

Python is useful for numerical estimates, exact symbolic forms, and checking units in repeated calculations.

import math

height = 40 * math.tan(math.radians(35))
round(height, 1)
28.0

Use exact SymPy values when the angle is special.

import sympy as sp

horizontal = 50 * sp.cos(sp.pi / 6)
vertical = 50 * sp.sin(sp.pi / 6)
sp.simplify(horizontal), sp.simplify(vertical)
(25*sqrt(3), 25)

Use edumath helpers to keep degree-based computations explicit.

from edumath.trigonometry import sine_cosine_tangent

sine, cosine, tangent = sine_cosine_tangent(35)
round(tangent, 4)
0.7002

A sinusoidal model can also be evaluated directly.

t = sp.symbols("t")
tide = 3 * sp.cos((sp.pi / 6) * t) + 5
sp.simplify(tide.subs(t, 6))

\(\displaystyle 2\)