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AI HL · 12.10

Geometry & Trigonometry

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12.10.11

The unit circle and trigonometric equations

12.10.11

The unit circle and trigonometric equations

Target: I can use the unit circle to define cos θ and sin θ, and find the coordinates of a point on a circle from its angle. (+5 more)

By the end of this section

Key points

The unit circle

  • The unit circle has centre O at the origin and radius 1. Measure an angle θ anticlockwise from the positive x-axis. The point P at that angle is \((\cos\theta,\sin\theta)\).
  • So cos θ is the x-coordinate and sin θ is the y-coordinate. On a circle of radius r the point is \((r\cos\theta,\;r\sin\theta)\).
  • Signs by quadrant: first, both positive. Second, sine positive and cosine negative. Third, both negative. Fourth, cosine positive and sine negative.
  • An angle bigger than 360° or negative takes you to the same point as the angle plus or minus whole turns.

Two identities

  • \(\cos^2\theta+\sin^2\theta=1\). It is Pythagoras in the right-angled triangle inside the unit circle.
  • \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). It is the gradient of the line OP.
  • To find cos θ from sin θ: \(\cos\theta=\pm\sqrt{1-\sin^2\theta}\). Choose the sign from the quadrant. Do not leave ±.
  • Divide an equation such as \(a\sin x=b\cos x\) by \(\cos x\) to get \(\tan x=\frac{b}{a}\).

Two angles with the same value

  • Same sine means the same height. \(\sin\theta=\sin(180^\circ-\theta)\), or \(\sin\theta=\sin(\pi-\theta)\) in radians.
  • Same cosine means the same x-coordinate. \(\cos\theta=\cos(360^\circ-\theta)\), or \(\cos\theta=\cos(2\pi-\theta)\).
  • The calculator gives one angle only: \(\sin^{-1}\) between −90° and 90°, \(\cos^{-1}\) between 0° and 180°. The second one is yours to find.
  • Tangent repeats every 180° (π radians): add 180° for the second solution.

Solving in an interval

  • The interval decides how many solutions there are. Sketch or graph first.
  • \(\sin bx\) and \(\cos bx\) repeat every \(\frac{360^\circ}{b}\) (or \(\frac{2\pi}{b}\)). A bigger b means more solutions in the same interval.
  • Check the mode (degrees or radians) matches the interval. A radian interval with the calculator in degrees gives wrong answers.
  • In a model, a time of 20 m height or a depth of 7 m is a graph and a horizontal line: the duration is the gap between two solutions.

On the calculator

Solving a trigonometric equation from the graph

  • Set the mode to degrees or radians to match the question.
  • Enter the left side as one graph and the right side as another, for example \(3\sin2x\) and \(1.8\).
  • Set the window to the interval in the question and the height of the graph.
  • Find each intersection. Count them: they should fit the period and the interval.
  • Write the equation, then the solutions to the accuracy asked.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A point on a circle

A point P lies on a circle with centre O at the origin and radius 5 cm. The line OP makes an angle of 130° with the positive x-axis, measured anticlockwise. Find the coordinates of P.

5 cm130°OP
Working
  1. The rule

    \(P=(r\cos\theta,\;r\sin\theta)\)

    On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate. A circle of radius r scales both by r.

  2. x-coordinate

    \(5\cos130^\circ=-3.214\)

    Negative: the point is to the left of the y-axis.

  3. y-coordinate

    \(5\sin130^\circ=3.830\)

    Positive: the point is above the x-axis.

  4. Answer

    \(P=(-3.21,\;3.83)\)

    Calculator in degree mode.

Video walkthrough coming
Cosine and tangent from the sine

The angle θ is obtuse and \(\sin\theta=\frac{7}{25}\). Find the exact values of (a) \(\cos\theta\), (b) \(\tan\theta\).

Working
  1. The identity

    \(\cos^2\theta+\sin^2\theta=1\)

    From Pythagoras in the right-angled triangle inside the unit circle.

  2. Square of cosine

    \(\cos^2\theta=1-\left(\frac{7}{25}\right)^2=\frac{625-49}{625}=\frac{576}{625}\)

  3. Square root

    \(\cos\theta=\pm\frac{24}{25}\)

    Two possible signs.

  4. The quadrant

    \(\theta\text{ is obtuse, so }\cos\theta=-\frac{24}{25}\)

    In the second quadrant the x-coordinate is negative.

  5. Tangent

    \(\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{7/25}{-24/25}=-\frac{7}{24}\)

    The 25s cancel.

Video walkthrough coming
The same sine, the same cosine

Given that sin 28° = 0.4695, write down another angle between 0° and 360° with the same sine. Given that cos 1.1 = 0.4536 (radians), write down another angle between 0 and 2π with the same cosine.

90180270360−1−0.50.51x°yy = sin x
Working
  1. Sine is the height

    sin θ = y-coordinate

    Two points on the circle are at the same height, one each side of the y-axis.

  2. Other angle, sine

    \(180^\circ-28^\circ=152^\circ\)

    In radians: π − θ.

  3. Cosine is the x-coordinate

    cos θ = x-coordinate

    Two points are at the same distance along the x-axis, one above and one below.

  4. Other angle, cosine

    \(2\pi-1.1=5.18\)

    In degrees: 360° − θ.

Video walkthrough coming
Solving in degrees

Solve for 0° ≤ x ≤ 360°: (a) \(\sin x=0.35\), (b) \(\cos x=-0.4\).

90180270360−1−0.50.51x°yy = cos x
Working
  1. (a) Calculator angle

    \(x=\sin^{-1}0.35=20.5^\circ\)

    From the calculator, degree mode.

  2. (a) Second angle

    \(x=180^\circ-20.5^\circ=159.5^\circ\)

    Sine: π − θ. Both are between 0° and 360°.

  3. (b) Calculator angle

    \(x=\cos^{-1}(-0.4)=113.6^\circ\)

    The calculator gives an angle between 0° and 180°.

  4. (b) Second angle

    \(x=360^\circ-113.6^\circ=246.4^\circ\)

    Cosine: 2π − θ.

  5. Answers

    \(x=20.5^\circ\text{ or }159.5^\circ;\;\;x=113.6^\circ\text{ or }246.4^\circ\)

    Check each by putting it back into the equation.

Video walkthrough coming
An equation in radians

Solve \(3\sin 2x=1.8\) for \(0\le x\le2\pi\).

123456−3−2−1123xyy = 3 sin 2x
Working
  1. Radian mode

    Set the calculator to radians

    The interval is in radians.

  2. Graph both sides

    \(y=3\sin2x\quad\text{and}\quad y=1.8\)

    Set the window to 0 ≤ x ≤ 2π.

  3. Intersections

    \(x=0.322,\;1.249,\;3.463,\;4.391\)

    From the calculator. The graph repeats every π, so there are two crossings in each π.

  4. Check

    \(0.322+\pi=3.463\)

    Adding the period π to the first solution gives the third.

  5. Answer

    \(x=0.322,\;1.249,\;3.463,\;4.391\)

    Four solutions, to 3 decimal places.

Video walkthrough coming
Using tan x = sin x ÷ cos x

Solve \(3\sin x=2\cos x\) for \(0^\circ\le x\le360^\circ\).

Working
  1. Divide by cos x

    \(\frac{\sin x}{\cos x}=\frac23\)

    Divide both sides by \(3\cos x\). Here \(\cos x\ne0\).

  2. Use tan

    \(\tan x=\frac23\)

    \(\tan x=\sin x\div\cos x\).

  3. Calculator angle

    \(x=\tan^{-1}\frac23=33.7^\circ\)

  4. Second angle

    \(x=33.7^\circ+180^\circ=213.7^\circ\)

    Tangent repeats every 180°, so add 180° and stay inside the interval.

  5. Answer

    \(x=33.7^\circ\text{ or }213.7^\circ\)

Video walkthrough coming
A Ferris wheel

A seat on a Ferris wheel is at its lowest point when t = 0, where t is the time in minutes. Its height above the ground in metres is \(h=14-12\cos\left(\frac{\pi t}{8}\right)\), for \(0\le t\le16\). (a) Find the height of the seat when t = 3. (b) Find the times when the seat is 19 m above the ground. (c) Find for how long in each turn the seat is more than 19 m above the ground.

246810121416481216202428t (min)h (m)h
Working
  1. (a) Height

    \(h(3)=14-12\cos\frac{3\pi}{8}=9.41\) m

    Radian mode. The angle is \(\frac{3\pi}{8}\) radians.

  2. (b) Equation

    \(14-12\cos\left(\frac{\pi t}{8}\right)=19\)

    Graph \(h\) and \(h=19\) for \(0\le t\le16\).

  3. (b) Intersections

    \(t=5.09\text{ and }t=10.91\) minutes

    From the calculator.

  4. (c) Duration

    \(10.906-5.094=5.81\) minutes

    The seat is above 19 m between the two times, where the graph is above the line.

Video walkthrough coming