PT1 - Trigonometric ratios of any angle

PT1 - Trigonometric ratios of any angle

Extend sine, cosine and tangent beyond right-angled triangles by following a point around a circle. You will find signed and exact values, then sketch and plot the three trigonometric graphs in degrees.

Sine and cosine as coordinates

A point travels round a circle of radius 11. At 30∘30^\circ its height is 12\frac12. Where is it at 150∘150^\circ, and why is its height still 12\frac12? A right-angled triangle alone cannot contain an angle of 150∘150^\circ, but a rotation can.

You already know

In a right-angled triangle, sine is opposite divided by hypotenuse, cosine is adjacent divided by hypotenuse and tangent is opposite divided by adjacent. Set your calculator to degrees for this course.

Start at (1,0)(1, 0). Rotate anticlockwise through an angle θ\theta; a negative angle means clockwise. If the new point is P(x,y)P(x, y), its horizontal coordinate is cos⁡θ\cos\theta and its vertical coordinate is sin⁡θ\sin\theta. In the first quadrant, dropping a perpendicular produces your familiar right-angled triangle with hypotenuse 11. The coordinate definition agrees with GCSE trigonometry and still works after the point passes either axis.

[DIAGRAM: asset_name: Sine and cosine round the unit circle; asset_slug: any-angle-unit-circle; caption: Angles of 30 degrees and 150 degrees have the same height but opposite horizontal coordinates; description: A unit circle centred at O has horizontal and vertical coordinate axes. Points P and Q are at rotations of 30 and 150 degrees. Blue radii join O to the points, sage dotted perpendiculars show a shared height of one half and opposite horizontal coordinates root 3 over 2 and minus root 3 over 2. Anticlockwise angle arcs show the rotations.]
Angles of 30 degrees and 150 degrees have the same height but opposite horizontal coordinates

Unit-circle definitions

On the circle of radius 11 centred at the origin, the point reached by an anticlockwise rotation of θ\theta from the positive horizontal axis has coordinates

P=(cos⁡θ,sin⁡θ)P=(\cos\theta, \sin\theta)

For example, 90∘90^\circ gives (0,1)(0, 1), so cos⁡90∘=0\cos90^\circ=0 and sin⁡90∘=1\sin90^\circ=1. This definition includes angles outside a single turn.

A complete turn returns the point to exactly the same place. So 30∘30^\circ, 390∘390^\circ and −330∘-330^\circ all have the same sine and cosine. This is a useful surprise: infinitely many different rotations can have exactly the same coordinates. The rotation records the journey; the trigonometric values record the destination.

Repeating sine and cosine

sin⁡(θ+360∘n)=sin⁡θcos⁡(θ+360∘n)=cos⁡θ\sin(\theta+360^\circ n)=\sin\theta \qquad \cos(\theta+360^\circ n)=\cos\theta

These hold for every angle θ\theta and every integer nn. A full turn does not change either coordinate.
Not on the formulae sheet: learn it.

The axes also tell you the signs. In quadrant I both coordinates are positive; in II the horizontal coordinate is negative; in III both are negative; in IV only the vertical coordinate is negative. Points on the axes belong to neither neighbouring quadrant.

Key idea

Sine measures signed height and cosine measures signed horizontal position. A negative value describes a direction, not a negative side length.

Use the coordinates to diagnose this claim rather than trying to draw an obtuse angle inside a right-angled triangle.

Signed and exact trigonometric values

Use a small acute reference angle to find the size of a ratio, then use the coordinates to decide its sign.

The reference angle is the acute angle between the terminal radius and the horizontal axis. For example, 150∘150^\circ has reference angle 30∘30^\circ; 240∘240^\circ has reference angle 60∘60^\circ. On an axis, read the coordinates directly instead. Tangent is the ratio of signed vertical to signed horizontal position, so it is positive in quadrants I and III and negative in II and IV.

Tangent from coordinates

tan⁡θ=yx\tan\theta=\frac{y}{x}

Here (x,y)=(cos⁡θ,sin⁡θ)(x, y)=(\cos\theta, \sin\theta) and x≠0x\ne0. Thus tangent is undefined at 90∘+180∘n90^\circ+180^\circ n, for integer nn. The right-angled ratio is on the formulae sheet; this coordinate extension is not.
Not on the formulae sheet: learn it.

The familiar exact values come from a right-angled isosceles triangle with sides 1,1,21, 1, \sqrt2, and half of an equilateral triangle with sides 1,3,21, \sqrt3, 2. Their ratios give this table; do not round them before using them.

AngleSineCosineTangent
0∘0^\circ001100
30∘30^\circ12\frac1232\frac{\sqrt3}{2}33\frac{\sqrt3}{3}
45∘45^\circ22\frac{\sqrt2}{2}22\frac{\sqrt2}{2}11
60∘60^\circ32\frac{\sqrt3}{2}12\frac123\sqrt3
90∘90^\circ1100Undefined

These exact values are not on the formulae sheet: learn them, or reconstruct them from the two triangles. Reflection in an axis preserves the reference triangle's lengths and changes only the appropriate coordinate sign.

Worked example 1: a quadrant II angle

Question. Find the exact values of sin⁡150∘\sin150^\circ, cos⁡150∘\cos150^\circ and tan⁡150∘\tan150^\circ.

Plan. Use reference angle 30∘30^\circ, then assign the signs from quadrant II.

180∘−150∘=30∘reference anglesin⁡150∘=sin⁡30∘=12positive heightcos⁡150∘=−cos⁡30∘=−32negative horizontal positiontan⁡150∘=1/2−3/2=−33divide the coordinates\begin{aligned} 180^\circ-150^\circ&=30^\circ &&\text{reference angle}\\ \sin150^\circ&=\sin30^\circ=\frac12 &&\text{positive height}\\ \cos150^\circ&=-\cos30^\circ=-\frac{\sqrt3}{2} &&\text{negative horizontal position}\\ \tan150^\circ&=\frac{1/2}{-\sqrt3/2}=-\frac{\sqrt3}{3} &&\text{divide the coordinates} \end{aligned}

Why decide signs separately? The reference triangle has positive lengths, whereas the circle has signed coordinates.

Answer. 12\frac12, −32-\frac{\sqrt3}{2} and −33-\frac{\sqrt3}{3}, respectively.

Check. A calculator in degree mode gives approximately 0.5000.500, −0.866-0.866 and −0.577-0.577, with the expected signs.

The next example changes only the direction of rotation. Reflecting the point in the horizontal axis changes the height's sign but keeps its horizontal position.

Worked example 2: a negative angle

Question. Find the exact values of sin⁡(−150∘)\sin(-150^\circ), cos⁡(−150∘)\cos(-150^\circ) and tan⁡(−150∘)\tan(-150^\circ).

Plan. Add one full turn to locate the point, then use reference angle 30∘30^\circ.

−150∘+360∘=210∘same destination210∘−180∘=30∘reference anglesin⁡(−150∘)=−12quadrant III heightcos⁡(−150∘)=−32quadrant III horizontal positiontan⁡(−150∘)=−1/2−3/2=33two negatives divide\begin{aligned} -150^\circ+360^\circ&=210^\circ &&\text{same destination}\\ 210^\circ-180^\circ&=30^\circ &&\text{reference angle}\\ \sin(-150^\circ)&=-\frac12 &&\text{quadrant III height}\\ \cos(-150^\circ)&=-\frac{\sqrt3}{2} &&\text{quadrant III horizontal position}\\ \tan(-150^\circ)&=\frac{-1/2}{-\sqrt3/2}=\frac{\sqrt3}{3} &&\text{two negatives divide} \end{aligned}

Why add 360∘360^\circ? It identifies the quadrant without changing the point or any ratio.

Answer. −12-\frac12, −32-\frac{\sqrt3}{2} and 33\frac{\sqrt3}{3}, respectively.

Check. Reflection from 150∘150^\circ reverses sine and tangent, while cosine stays negative.

An undefined value must stay undefined, however much a calculator encourages you to type it.

Watch out

Wrong: tan⁡90∘=0\tan90^\circ=0, because the horizontal coordinate is zero. Right: tan⁡90∘=1/0\tan90^\circ=1/0 is undefined. By contrast, tan⁡180∘=0/(−1)=0\tan180^\circ=0/(-1)=0.

Try the same method with a different reference angle.

Sketching and plotting sine and cosine

A trigonometric graph records one coordinate as the angle changes, so its waves come directly from circular motion.

For sine, follow the height: 0,1,0,−1,00, 1, 0, -1, 0 at quarter-turn steps from 0∘0^\circ to 360∘360^\circ. For cosine, follow horizontal position: 1,0,−1,0,11, 0, -1, 0, 1. Both repeat every 360∘360^\circ, stay between −1-1 and 11, and continue smoothly to negative angles and beyond one revolution.

Period and amplitude

The period is the smallest positive horizontal shift that repeats the whole graph. The amplitude of a sine or cosine graph is half the distance between its greatest and least values. Both y=sin⁡θy=\sin\theta and y=cos⁡θy=\cos\theta have period 360∘360^\circ and amplitude 11.

A sketch shows the main features with labelled key coordinates. A plot uses calculated points on a scale: use enough points to capture the curve, then join them smoothly, rather than with straight chords. For these graphs, quarter-turn points alone give the outline; extra 30∘30^\circ or 45∘45^\circ points make a more accurate plot. Their periods, amplitudes and coordinate patterns are not on the formulae sheet: learn them.

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