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Trigonometry

Why this chapter

Why this chapter exists

You may go on with the physics now. When a lesson needs this tool, it will tell you and bring you here.

A fielder throws at an angle, a block rests on a slope and a fan turns. The mains voltage rises and falls, two tuning forks beat and a pendulum clock keeps time. Each of these leans on angles. The physics must split a velocity along two directions and measure a turn in radians. It must write a quantity that repeats, add two sines and solve a triangle of forces. This chapter teaches exactly those tools, each with a physics example from the book and its own numbers.

You will learn

By the end of this chapter you will

Split a vector at an angle into \(u\cos\theta\) and \(u\sin\theta\), with the standard angles, the 3-4-5 triangle and the signs in the four quadrants.
Measure angles in radians, convert them to degrees and back, and use \(s = r\theta\) and \(v = \omega r\).
Read the amplitude, the period and the phase of \(y = A\sin(\omega t + \phi)\), and say how a phase slides the curve.
Use \(\sin^2\theta + \cos^2\theta = 1\), the sum and double-angle formulas and the sum-to-product identities.
Solve any triangle with the sine rule and the cosine rule, including the triangle of forces and the resultant of two vectors.
Replace \(\sin\theta\) by \(\theta\) when it is safe, say when it is not, and find an angle back from its sine.

Before you begin

You should know Pythagoras' theorem, what similar triangles are, and how to rearrange a simple formula. Each lesson stands alone and starts from a physics situation, so you may open any of them from a physics chapter. The last lesson borrows one line from the chapter Differentiation, and it says so where it does.

Already know this?

Lessons

Sine, Cosine and Tangent

The three ratios of a right triangle and how they split a velocity or a force into two parts. The standard angles from \(0°\) to \(90°\), with the 3-4-5 triangle, and the unit circle that carries the ratios to every angle with their signs in the four quadrants.

Sine cosine and tangent of an angleStandard angles and the 3 4 5 triangleThe unit circle and the four quadrants

16 min0%

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Radians and Arc Length

The radian as the arc length divided by the radius, \(\theta = s/r\), so that a full turn is \(2\pi\) radians. Converting between degrees and radians, and angular speed in radians per second, with \(v = \omega r\) for the tip of a ceiling fan.

Radians and arc lengthConverting degrees and radiansThe angle does not depend on the radiusAngular speed in radians per second

13 min0%

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Amplitude, Period and Phase

The sine and the cosine as functions of time, \(y = A\sin(\omega t + \phi)\). The amplitude \(A\), the angular frequency \(\omega\) with the period \(T = 2\pi/\omega\), and the phase \(\phi\) that slides the whole curve along the time axis.

The graphs of sine and cosineAmplitude period and phaseThe phase slides the curve

15 min0%

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Identities Physics Uses

The identity \(\sin^2\theta + \cos^2\theta = 1\), the sum and double-angle formulas behind the range \(u^2\sin 2\theta/g\), and the sum-to-product identities that turn two close notes into beats and two opposite waves into a standing wave.

The pythagorean identityDouble angle and sum formulasSum to product identities

16 min0%

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The Sine Rule, the Cosine Rule and the Triangle of Forces

Two rules for any triangle, \(a/\sin A = b/\sin B = c/\sin C\) and \(c^2 = a^2 + b^2 - 2ab\cos C\). They solve the triangle of forces of a hanging load, which is Lami's theorem, and they give the resultant of two vectors at any angle.

Sine rule and cosine ruleThe triangle of forces and lamis theoremThe resultant of two vectors

15 min0%

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Small Angles and Inverse Functions

For a small angle in radians, \(\sin\theta \approx \tan\theta \approx \theta\) and \(\cos\theta \approx 1 - \theta^2/2\). How fast the error grows and where it passes one per cent, and the inverse functions that give an angle back from its sine, as when the angle of projection is found from a range.

Small angle approximationsThe cosine of a small angleFinding an angle from its sine

15 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

Glossary

The key words of this chapter, with the meaning this chapter gives them.

amplitude: The largest value of a quantity that varies as \(A\sin(\omega t + \phi)\); the quantity swings between \(+A\) and \(-A\).
angular frequency: The rate \(\omega\), in rad/s, at which the angle inside a sinusoid grows; it fixes the period \(T = 2\pi/\omega\).
angular speed: The angle a turning body sweeps in each second, \(\omega\), in rad/s. A point at distance \(r\) from the axis then moves at \(v = \omega r\).
component: The part of a vector along a chosen direction. A vector of size \(u\) at an angle \(\theta\) to an axis has the component \(u\cos\theta\) along that axis.
hypotenuse: The longest side of a right triangle, the side across from the right angle.
identity: An equation between trigonometric ratios that holds for every value of the angle, such as \(\sin^2\theta + \cos^2\theta = 1\).
inverse sine: The angle whose sine is a given number, written \(\sin^{-1}x\). A calculator gives the value between \(-90°\) and \(90°\); the angle \(180°\) minus it has the same sine.
Lami's theorem: For three forces in equilibrium at a point, each force divided by the sine of the angle between the other two gives the same number. It is the sine rule of the triangle of forces.
period: The time \(T\) after which a repeating quantity takes the same value again, in the same direction. For a sinusoid, \(T = 2\pi/\omega\).
phase: The angle \(\omega t + \phi\) inside a sinusoid, in radians. Its value at \(t = 0\), the constant \(\phi\), sets where the cycle starts.
quadrant: One of the four quarters into which the two axes cut the plane, numbered anticlockwise from the one where \(x\) and \(y\) are both positive.
radian: The unit of angle in which an angle is the arc it cuts off divided by the radius, \(\theta = s/r\). A full turn is \(2\pi\) radians, and 1 rad is 57.3°.
resultant: The single vector equal to the sum of two or more vectors. For \(P\) and \(Q\) at an angle \(\theta\), \(R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta}\).
triangle of forces: The closed triangle made by three forces in equilibrium when they are drawn head to tail. Its sides are the sizes of the forces.
unit circle: A circle of radius 1 with its centre at the origin. The point at the angle \(\theta\) on it is \((\cos\theta, \sin\theta)\).