Vectors and Components
A free Trigonometry lesson from the “Applications of Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.
Vector components use cosine for the horizontal part and sine for the vertical part when the angle is measured from the positive x-axis. In this lesson, the goal is to split a vector into horizontal and vertical components. Prerequisite check: Algebra II or College Algebra foundations. Example 1: a 45 degree angle of elevation from 20 ft away gives height 20 ft because tan(45 degrees) = 1. Example 2: a vector of length 10 at 30 degrees has horizontal component 10cos(30 degrees) and vertical component 10sin(30 degrees). A common mistake is using a formula before identifying the included angle or opposite side; the safer habit is to draw and label the triangle, vector, or cycle before writing an equation.
What you'll learn
- Split a vector into horizontal and vertical components
- choose the trig tool that matches the diagram and known information
- Explain why applications show how trig supports STEM and design problems
Worked example
Problem. Example 1 Foundation: A daylight model repeats once every year. What graph feature describes one full repeat?
- The period is the time for one full cycle.
- A yearly pattern has period one year.
- Amplitude and midline describe vertical features.
Answer: period
Practice problems
1. Practice 1 Foundation: A ladder makes a 60 degree angle with the ground and reaches 12 ft up a wall. Which equation can find the ladder length L?
Choices: sin(60 degrees) = 12/L · cos(60 degrees) = 12/L · tan(60 degrees) = L/12 · sin(60 degrees) = L/12
Show solution
- Warm-up: First identify exactly what the question is asking: Practice 1 Foundation: A ladder makes a 60 degree angle with the ground and reaches 12 ft up a wall. Which equation can find the ladder length L?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- The height is opposite the 60 degree angle.
- The ladder is the hypotenuse.
- Sine uses opposite over hypotenuse.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: sin(60 degrees) = 12/L
2. Practice 2 Setup: Which formula is the Law of Sines?
Choices: a/sin(A) = b/sin(B) = c/sin(C) · c^2 = a^2 + b^2 - 2ab cos(C) · Area = (1/2)ab sin(C) · tan(theta) = opposite/adjacent
Show solution
- The Law of Sines matches each side with the sine of its opposite angle.
- It is useful when angle-side pairs are known.
- Do not use it as the area formula.
Answer: a/sin(A) = b/sin(B) = c/sin(C)
3. Practice 3 Meaning: Which formula is the Law of Cosines?
Choices: c^2 = a^2 + b^2 - 2ab cos(C) · a/sin(A) = b/sin(B) · Area = (1/2)ab sin(C) · tan(theta) = sin(theta)/cos(theta)
Show solution
- The Law of Cosines relates three sides and an included angle.
- It extends the Pythagorean theorem.
- Use it when the included angle is known.
Answer: c^2 = a^2 + b^2 - 2ab cos(C)
4. Practice 4 Method: Two sides of a triangle are 8 and 10 with included angle 30 degrees. Which area setup is correct?
Choices: Area = (1/2)(8)(10)sin(30 degrees) · Area = 8 + 10 + 30 · Area = (8)(10)cos(30 degrees) · Area = (1/2)(8)(10)tan(30 degrees)
Show solution
- The trig area formula is (1/2)ab sin(C).
- Use the two sides and the included angle.
- Here that is (1/2)(8)(10)sin(30 degrees).
Answer: Area = (1/2)(8)(10)sin(30 degrees)
5. Practice 5 Reasoning: A vector has length 10 at 30 degrees above the positive x-axis. What is its vertical component?
Choices: 10sin(30 degrees) · 10cos(30 degrees) · 10tan(30 degrees) · 10/sin(30 degrees)
Show solution
- Vertical component uses sine when the angle is measured from the x-axis.
- Horizontal component uses cosine.
- So the vertical component is 10sin(30 degrees).
Answer: 10sin(30 degrees)
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