3.3 - Sine, cosine and tangent graphs

3.3 - Sine, cosine and tangent graphs

Sine, cosine and tangent keep working after an angle passes a full turn. You will draw their graphs in degrees, recognise their repeating patterns and find all the angles that give a chosen output. Keep your calculator in degree mode throughout.

The sine graph (Higher)

How can an angle of 450∘450^\circ give the same sine as 90∘90^\circ? Imagine a point turning round a circle of radius 11. After one complete turn it is back in the same place, so another quarter turn reaches the same height as the first quarter turn.

You already know

In a right-angled triangle, sine is opposite divided by hypotenuse. Coordinates describe a point's horizontal and vertical position. A full turn is 360∘360^\circ.

The circle picture extends the triangle ratio: sine is the vertical coordinate of the turning point. It can be negative below the horizontal axis. The output is a ratio, so it has no length unit.

Period

A graph's period is the smallest positive input change that repeats the whole graph. Sine has period 360∘360^\circ. Its domain is all real degree angles and its range is −1≤y≤1-1\leq y\leq1.

Start a drawing with the values below. Label a regular degree scale, plot the points, then join them with a smooth wave. Use extra calculator values between them if you need help with the shape. Peaks and troughs are rounded, with horizontal tangents.

xx in degrees009090180180270270360360
sin⁡x\sin x001100−1-100

[DIAGRAM: asset_slug: m052_sine; caption: Sine repeats every 360 degrees, including negative and larger angles; description: Exact degree-mode y=sin x from negative 360 to 720 with outputs between negative 1 and 1 and key points every 90 degrees; file: m052_sine.png; recommended_method: matplotlib]
Sine repeats every 360 degrees, including negative and larger angles

The picture repeats to the left as well as the right. Sine is positive in the upper half-turn and negative in the lower half-turn.

Sine cycles

sin⁡(x+360∘)=sin⁡x,sin⁡(−x)=−sin⁡x\sin(x+360^\circ)=\sin x,\qquad \sin(-x)=-\sin x

These hold for every real degree angle. They are not in the Higher formulae sheet: learn them. This lesson is Higher only.

Use the cycle to turn a large angle into a familiar one.

Worked example 1: another full turn

Question. Find sin⁡450∘\sin450^\circ.

Plan. Remove one full turn, then use a key point.

450∘=90∘+360∘separate a full turnsin⁡450∘=sin⁡90∘repeat the cycle=1read the peak\begin{aligned} 450^\circ&=90^\circ+360^\circ &&\text{separate a full turn}\\ \sin450^\circ&=\sin90^\circ &&\text{repeat the cycle}\\ &=1 &&\text{read the peak} \end{aligned}

Why this step? A full turn returns the circle point to the same height.

Answer. sin⁡450∘=1\sin450^\circ=1.

Check. The graph has a peak at 450∘450^\circ, and sine cannot exceed 11.

Change only the direction of the angle next. Turning backwards reverses the vertical coordinate for sine.

Worked example 2: a negative input

Question. Find sin⁡(−450∘)\sin(-450^\circ).

Plan. Use the sign relationship, then the same full-turn reduction.

sin⁡(−450∘)=−sin⁡450∘reverse the angle=−sin⁡90∘remove a full turn=−1use the key value\begin{aligned} \sin(-450^\circ)&=-\sin450^\circ &&\text{reverse the angle}\\ &=-\sin90^\circ &&\text{remove a full turn}\\ &=-1 &&\text{use the key value} \end{aligned}

Why this step? Sine measures height, which changes sign when the circle point is reflected below the horizontal axis.

Answer. −1-1.

Check. Adding 720∘720^\circ gives 270∘270^\circ, where the graph has value −1-1.

A useful surprising fact is that sine can distinguish opposite directions even when the angle sizes are equal: sin⁡30∘=12\sin30^\circ=\tfrac12, but sin⁡(−30∘)=−12\sin(-30^\circ)=-\tfrac12.

Watch out

The line sin⁡450∘=sin⁡45∘\sin450^\circ=\sin45^\circ is wrong: removing a zero is not removing a turn. Subtract 360∘360^\circ, giving 90∘90^\circ, not 45∘45^\circ.

Try the same idea with a different negative input.

The cosine graph (Higher)

Cosine gives the horizontal coordinate on the same unit circle. At zero angle the point is at the rightmost edge, so cosine starts at 11, while sine starts at 00.

Cosine graph

The graph y=cos⁡xy=\cos x has domain all real degree angles, range −1≤y≤1-1\leq y\leq1 and period 360∘360^\circ. Its symmetry about the vertical axis means cos⁡(−x)=cos⁡x\cos(-x)=\cos x.

Plot these key points, join smoothly and repeat the cycle. The first minimum is at 180∘180^\circ; the first two zeros are at 90∘90^\circ and 270∘270^\circ.

xx in degrees009090180180270270360360
cos⁡x\cos x1100−1-10011

[DIAGRAM: asset_slug: m052_cosine; caption: Cosine starts at a maximum and is symmetric about the vertical axis; description: Exact y=cos x in degrees from negative 360 to 720, period 360 and range negative 1 to 1; file: m052_cosine.png; recommended_method: matplotlib]
Cosine starts at a maximum and is symmetric about the vertical axis

Cosine's cycle and symmetry facts are not in the Higher formulae sheet: learn them. Negative input does not automatically mean negative output.

Worked example 3: cosine at a negative angle

Question. Find cos⁡(−120∘)\cos(-120^\circ).

Plan. Use cosine's symmetry, then a familiar circle position.

cos⁡(−120∘)=cos⁡120∘vertical-axis symmetry=−12horizontal coordinate\begin{aligned} \cos(-120^\circ)&=\cos120^\circ &&\text{vertical-axis symmetry}\\ &=-\tfrac12 &&\text{horizontal coordinate} \end{aligned}

Why this step? Opposite rotations have the same horizontal coordinate.

Answer. −12-\tfrac12.

Check. Degree-mode calculation gives −0.5-0.5, within the permitted range.

Keep the same angle size within a cycle, but add two full turns to the input.

Worked example 4: a negative angle beyond a cycle

Question. Find cos⁡(−840∘)\cos(-840^\circ).

Plan. Add complete turns until the input becomes familiar.

−840∘+720∘=−120∘add two turnscos⁡(−840∘)=cos⁡(−120∘)same cycle position=−12use symmetry\begin{aligned} -840^\circ+720^\circ&=-120^\circ &&\text{add two turns}\\ \cos(-840^\circ)&=\cos(-120^\circ) &&\text{same cycle position}\\ &=-\tfrac12 &&\text{use symmetry} \end{aligned}

Why this step? Periodicity changes the input while leaving the output unchanged.

Answer. −12-\tfrac12.

Check. Adding a further 360∘360^\circ gives 240∘240^\circ, another point with cosine −12-\tfrac12.

Both examples have negative outputs because their circle points are on the left, not because their inputs are negative.

Watch out

cos⁡(−120∘)=−cos⁡120∘\cos(-120^\circ)=-\cos120^\circ incorrectly uses sine's sign rule. Cosine is symmetric: cos⁡(−120∘)=cos⁡120∘=−12\cos(-120^\circ)=\cos120^\circ=-\tfrac12.

Check a new cycle position and its zeros.

Continue this lesson.

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