Angles of Any Magnitude

Mark the points on the unit circle at $60°$ and at $120°$. Without calculating anything, compare their heights and compare their horizontal positions. Which of the two coordinates is the same for both points, and which has changed?

Any angle's value has two parts. The sign comes from which quadrant the point lands in, since that fixes the signs of its coordinates. The size comes from the acute angle between the radius and the horizontal axis, which is called the related angle.

The point at angle $\theta$ is $(\cos\theta, \sin\theta)$, so the signs of the two ratios are simply the signs of the two coordinates.

First quadrant, $0°$ to $90°$: both coordinates positive, so $\sin\theta > 0$ and $\cos\theta > 0$. Their quotient is positive too, so all three ratios are positive.

Second quadrant, $90°$ to $180°$: first coordinate negative, second positive. So $\cos\theta < 0$, $\sin\theta > 0$, and $\tan\theta < 0$, since a positive divided by a negative is negative. Only sine is positive.

Third quadrant, $180°$ to $270°$: both negative. So sine and cosine are both negative, but their quotient is positive. Only tangent is positive.

Fourth quadrant, $270°$ to $360°$: first positive, second negative. Only cosine is positive.

Reading the positive one in each quadrant anticlockwise gives All, Sin, Tan, Cos, which is the usual mnemonic. It is worth deriving rather than memorising: if you can state the coordinate signs of a quadrant, you can rebuild the whole diagram in a few seconds and you will not misremember it.

The sign is settled. The size comes from a triangle you already know.

For any angle, drop a perpendicular from its point on the circle to the horizontal axis. The acute angle formed at the origin between the radius and the horizontal axis is the related angle.

The resulting right-angled triangle has the same shape wherever the point is, so the sizes of its sides, and hence the sizes of the ratios, are those of the related angle. Only the signs differ.

How to find it:

second quadrant: related angle $= 180° - \theta$;
third quadrant: $\theta - 180°$;
fourth quadrant: $360° - \theta$.

So for $\theta = 150°$, the related angle is $180° - 150° = 30°$. The point is in the second quadrant, where sine is positive, so

$$\sin 150° = +\sin 30° = \tfrac{1}{2}$$

and cosine is negative there, so $\cos 150° = -\cos 30° = -\tfrac{\sqrt{3}}{2}$.

Always measure to the horizontal axis, never the vertical. Measuring to the vertical gives the complementary angle and swaps sine with cosine, which is a different relationship and a common source of error.

Negative angles are measured clockwise from the positive horizontal axis. Turning $-60°$ lands in the fourth quadrant, at the same place as turning $+300°$.

The point at $-\theta$ is the mirror image, in the horizontal axis, of the point at $\theta$: the same horizontal position, the opposite height. So

$$\cos(-\theta) = \cos\theta \qquad \text{and} \qquad \sin(-\theta) = -\sin\theta$$

and consequently $\tan(-\theta) = -\tan\theta$.

Extra turns change nothing. Adding $360°$ takes the point once round the circle and back to where it started, so

$$\sin(\theta + 360°) = \sin\theta$$

and the same for cosine and tangent. Angles differing by a whole number of turns are called coterminal, and they share every trigonometric value.

This gives a standard first move for any awkward angle: add or subtract full turns until the angle lies between $0°$ and $360°$, then use the quadrant and related angle as usual. For $\theta = 780°$: subtracting two full turns gives $780° - 720° = 60°$, so $\sin 780° = \sin 60° = \tfrac{\sqrt{3}}{2}$.

Evaluating any exact ratio takes four steps.

1. Reduce to a standard angle. Add or subtract multiples of $360°$ until the angle is between $0°$ and $360°$.
2. Identify the quadrant, and hence the sign.
3. Find the related acute angle, measured to the horizontal axis.
4. Combine. Take the ratio of the related angle and attach the sign.

Two worked instances:

$\cos 210°$. Already between $0°$ and $360°$. It is in the third quadrant, where cosine is negative. The related angle is $210° - 180° = 30°$. So $\cos 210° = -\cos 30° = -\dfrac{\sqrt{3}}{2}$.

$\tan(-135°)$. Adding $360°$ gives $225°$, in the third quadrant, where tangent is positive. The related angle is $225° - 180° = 45°$. So $\tan(-135°) = +\tan 45° = 1$.

The order matters. Deciding the sign before finding the related angle keeps the two decisions separate, and separating them is what stops one error from causing another.

Five items on angles beyond the first quadrant. Work each one, then reveal the answer.

1. 1 In which quadrant is $200°$, and what is the sign of $\cos 200°$?

   Show answer
   Between $180°$ and $270°$; both coordinates are negative there.Third quadrant; negative

2. 2 Find the related angle for $310°$.

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   Fourth quadrant: $360° - \theta$.$50°$

3. 3 Find $\sin 150°$ exactly.

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   Second quadrant, sine positive, related angle $30°$.$\tfrac{1}{2}$

4. 4 Find $\cos(-45°)$ exactly.

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   Cosine is unchanged by a sign change in the angle.$\tfrac{1}{\sqrt{2}}$

5. 5 Simplify $\sin 400°$ to a first-quadrant ratio.

   Show answer
   Subtract a full turn.$\sin 40°$