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Right-Triangle Trigonometry and Vector Components

A vector at an angle is the hypotenuse of a right triangle.

Practice 1: Creating RepresentationsPractice 2: Mathematical Routines

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Question set for this topic

Part 1 · Hook

Why this matters

A kicked ball does two things at once: it travels across the field and it rises and falls. Trigonometry splits its velocity into those two parts, the horizontal and the vertical, so each can be handled with the one-line motion you already know.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Solve y = A sin θ for A.

  1. A = y / sin θ
  2. A = y sin θ
  3. A = sin θ / y
  4. A = y − sin θ
Show the answer

Divide both sides by sin θ.

  • Correct: A = y / sin θ:
  • A = y sin θ:
  • A = sin θ / y:
  • A = y − sin θ:

2. Displacements of +5.0 m and −2.0 m along a line add to

  1. +3.0 m
  2. +7.0 m
  3. −3.0 m
  4. −7.0 m
Show the answer

Add the signed values.

  • Correct: +3.0 m:
  • +7.0 m:
  • −3.0 m:
  • −7.0 m:

Part 4 · See it

See it first

Left: a soccer ball kicked from the grass with velocity 15 m/s at 37 degrees above the horizontal. The velocity arrow is the hypotenuse of a right triangle; its horizontal component, v x = v cos theta = 15 cos 37 degrees = 12 m/s, runs along the ground, and its vertical component, v y = v sin theta = 15 sin 37 degrees = 9.0 m/s, runs straight up. Right: a right triangle with angle theta, labeled hypotenuse, adjacent and opposite sides, with sin = opposite over hypotenuse, cos = adjacent over hypotenuse, tan = opposite over adjacent, the Pythagorean theorem, and a reminder to set the calculator to degree mode.
A velocity is the hypotenuse of a right triangle whose legs are its horizontal and vertical components. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. A vector points at an angle to the axesit forms the hypotenuse of a right triangle whose legs lie along the axes
  2. You know the magnitude and the anglethe leg adjacent to the angle is A cos θ and the leg opposite it is A sin θ
  3. You know both componentsthe magnitude is the square root of (x-component² + y-component²), and tan θ = y-component ÷ x-component
  4. You add vectors that are not along one lineyou add their x-components and their y-components separately

Part 6 · Key ideas

Key ideas

  • SOH CAH TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
  • Components: the side adjacent to the given angle is A cos θ; the side opposite it is A sin θ. Check which axis the angle is measured from.
  • Back to the vector: magnitude = √(Aₓ² + Ay²); angle = tan⁻¹(Ay/Aₓ).
  • Set your calculator to degree mode. The sheet lists 37° and 53° (the 3-4-5 triangle): sin 37° ≈ 0.60, cos 37° ≈ 0.80.
  • Add vectors tip to tail, or by adding x-components and y-components separately.

Part 7 · Misconception

A common mistake

The wrong idea: The x-component always uses cosine and the y-component always uses sine.

What actually happens: Cosine goes with the side adjacent to the angle you are given. If the angle is measured from the vertical (or from the ramp), the horizontal component uses sine.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. A ball is kicked at 20. m/s at 30.° above the horizontal. What is the horizontal component of its velocity? (Degree mode.)

Type a number in m/s.

Show the answer

The horizontal side is adjacent to the angle: vₓ = v cos θ = (20. m/s)(cos 30°) = 17 m/s.

  • Answer: 17 m/s

2. A hiker's displacement is 400 m at 53° north of east. What is the north component of the displacement? (Degree mode.)

Type a number in m.

Show the answer

North is the side opposite the 53° angle (measured from east): d(north) = d sin θ = 400 m × sin 53° = 320 m (2 significant figures; 3.2 × 10² m).

  • Answer: 320 m

3. A velocity has components vₓ = 9.0 m/s and vy = 12 m/s. What is its magnitude?

Type a number in m/s.

Show the answer

The components are the legs of a right triangle and the velocity is the hypotenuse: v = √(9.0² + 12²) = √225 = 15 m/s.

  • Answer: 15 m/s

4. A vector of magnitude A makes an angle θ with the vertical (y) axis. Which expression gives its horizontal (x) component?

  1. A cos θ
  2. A tan θ
  3. A sin θ
  4. A / cos θ
Show the answer

Do not memorize "x uses cosine". Find the side adjacent to the angle you are given: it gets cosine; the opposite side gets sine.

  • A cos θ: Cosine gives the side adjacent to θ, which here is the vertical side.
  • A tan θ: Tangent is a ratio of the two legs, not a leg divided by the hypotenuse.
  • Correct: A sin θ: Right: measured from the vertical, the horizontal side is opposite θ, so it uses sine.
  • A / cos θ: A component is shorter than the vector; dividing by cos θ makes it longer.

5. A student computes cos 60 on her calculator and gets −0.952. What went wrong?

  1. Cosine of 60° is negative
  2. The calculator is not in degree mode
  3. She should have used sine
  4. She should type the angle as 0.60
Show the answer

Physics 1 angles are in degrees. Set the calculator to degree mode before any trig: cos 60° = 0.500.

  • Cosine of 60° is negative: Cosine is positive for angles from 0° to 90°.
  • Correct: The calculator is not in degree mode: Right: it read 60 in another angle unit and gave −0.952. In degree mode, cos 60° = 0.500.
  • She should have used sine: Changing the function does not fix a calculator in the wrong angle mode.
  • She should type the angle as 0.60: Angles in degrees are entered as written; the mode is the problem.

6. Two displacements are added tip to tail: 3.0 m east, then 4.0 m north. Which describes the total displacement?

  1. 7.0 m, north of east
  2. 1.0 m, north
  3. 5.0 m, due north
  4. 5.0 m, north of east
Show the answer

Tip to tail: draw the second vector starting at the tip of the first; the resultant goes from the first tail to the last tip. Here it is a 3-4-5 triangle.

  • 7.0 m, north of east: Perpendicular displacements do not add as plain numbers; 7.0 m is the distance walked.
  • 1.0 m, north: Subtracting works only for vectors along the same line in opposite directions.
  • 5.0 m, due north: The resultant points from the start to the end, which is north of east, not due north.
  • Correct: 5.0 m, north of east: Right: the two legs are perpendicular, so the resultant is the hypotenuse √(3.0² + 4.0²) = 5.0 m, pointing between east and north.

Part 9 · Summary

Summary

A vector at an angle is the hypotenuse of a right triangle. The component adjacent to the given angle is A cos θ and the opposite one is A sin θ; the magnitude is the square root of the sum of the squared components. Add vectors by components, with the calculator in degree mode.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections