Exact values

In this chapter we focus on memorizing and using exact trigonometric values that come from the unit circle, without relying on a calculator. We assume you already know what the unit circle is and how sine and cosine relate to points on it.

We will concentrate on a small, very important set of angles and learn to express their sine, cosine, and tangent values using simple square roots and fractions.

The “special” angles

The angles whose exact values are used most often are:

• In degrees: $0^\circ,\; 30^\circ,\; 45^\circ,\; 60^\circ,\; 90^\circ$
• In radians: $0,\; \dfrac{\pi}{6},\; \dfrac{\pi}{4},\; \dfrac{\pi}{3},\; \dfrac{\pi}{2}$

On the unit circle, each of these angles corresponds to a point $(\cos\theta,\sin\theta)$, so knowing these points gives you the exact values of $\sin\theta$ and $\cos\theta$. Tangent is then
$$
\tan\theta = \frac{\sin\theta}{\cos\theta}
$$
whenever $\cos\theta\neq 0$.

Key exact values in the first quadrant

In the first quadrant (angles between $0$ and $90^\circ$ or $0$ and $\dfrac{\pi}{2}$), all sine and cosine values are positive.

Here are the standard exact values:

• $\theta = 0^\circ$ or $0$:
• $\cos 0 = 1$
• $\sin 0 = 0$
• $\tan 0 = 0$
• $\theta = 30^\circ$ or $\dfrac{\pi}{6}$:
• $\cos\dfrac{\pi}{6} = \dfrac{\sqrt{3}}{2}$
• $\sin\dfrac{\pi}{6} = \dfrac{1}{2}$
• $\tan\dfrac{\pi}{6} = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$
• $\theta = 45^\circ$ or $\dfrac{\pi}{4}$:
• $\cos\dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$
• $\sin\dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$
• $\tan\dfrac{\pi}{4} = 1$
• $\theta = 60^\circ$ or $\dfrac{\pi}{3}$:
• $\cos\dfrac{\pi}{3} = \dfrac{1}{2}$
• $\sin\dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}$
• $\tan\dfrac{\pi}{3} = \sqrt{3}$
• $\theta = 90^\circ$ or $\dfrac{\pi}{2}$:
• $\cos\dfrac{\pi}{2} = 0$
• $\sin\dfrac{\pi}{2} = 1$
• $\tan\dfrac{\pi}{2}$ is undefined (division by zero)

These five angles and their values form the core “exact values” you must know. All other special-angle values on the unit circle can be obtained by using symmetries and sign changes.

A memory aid for $\sin$ and $\cos$ in the first quadrant

A common pattern helps you remember the sine and cosine values for $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$:

For sine in the first quadrant:
$$
\sin 0^\circ = \sqrt{\frac{0}{4}},\quad
\sin 30^\circ = \sqrt{\frac{1}{4}},\quad
\sin 45^\circ = \sqrt{\frac{2}{4}},\quad
\sin 60^\circ = \sqrt{\frac{3}{4}},\quad
\sin 90^\circ = \sqrt{\frac{4}{4}}
$$

which simplifies to
$$
0,\; \frac{1}{2},\; \frac{\sqrt{2}}{2},\; \frac{\sqrt{3}}{2},\; 1
$$

Cosine follows the same pattern but in reverse order:
$$
\cos 0^\circ = 1,\;
\cos 30^\circ = \frac{\sqrt{3}}{2},\;
\cos 45^\circ = \frac{\sqrt{2}}{2},\;
\cos 60^\circ = \frac{1}{2},\;
\cos 90^\circ = 0
$$

Once you know $\sin\theta$ and $\cos\theta$, you can always get
$$
\tan\theta = \frac{\sin\theta}{\cos\theta}
$$
for the angles where it is defined.

Using quadrants for signs

To find exact values for angles beyond the first quadrant (between $0$ and $2\pi$), you do two things:

1. Use a reference angle in the first quadrant (an angle with the same “shape” relative to the $x$–axis).
2. Apply the correct sign based on the quadrant.

You do not need to memorize new square-root values for other quadrants: the numerical parts stay the same as the reference angle; only signs may change.

The sign pattern is often remembered by “ASTC” around the circle (starting in the first quadrant and going counterclockwise):

• Quadrant I: All of $\sin$, $\cos$, $\tan$ are positive.
• Quadrant II: Sine positive (cosine and tangent negative).
• Quadrant III: Tangent positive (sine and cosine negative).
• Quadrant IV: Cosine positive (sine and tangent negative).

Angles in each quadrant will be handled more generally in the “Unit Circle” parent chapter; here the emphasis is that once you know the first-quadrant exact values, you immediately know the values in other quadrants up to a sign.

For instance:

• $\theta = 150^\circ$ (which is $180^\circ - 30^\circ$) has reference angle $30^\circ$, is in Quadrant II, so:
• $\sin 150^\circ = \sin 30^\circ = \dfrac{1}{2}$ (positive in Quadrant II)
• $\cos 150^\circ = -\cos 30^\circ = -\dfrac{\sqrt{3}}{2}$
• $\tan 150^\circ = -\tan 30^\circ = -\dfrac{\sqrt{3}}{3}$

The numerical parts are exactly the same as at $30^\circ$; only the signs changed.

Common exact values around the circle

By combining the first-quadrant values with the sign rules, you can list exact values at standard angles throughout the circle. Here are some frequently used ones (in degrees and radians), written in terms of sine and cosine:

• $0^\circ$ or $0$: $(\cos\theta,\sin\theta) = (1,0)$
• $30^\circ$ or $\dfrac{\pi}{6}$: $\left(\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right)$
• $45^\circ$ or $\dfrac{\pi}{4}$: $\left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)$
• $60^\circ$ or $\dfrac{\pi}{3}$: $\left(\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right)$
• $90^\circ$ or $\dfrac{\pi}{2}$: $(0,1)$
• $120^\circ$ or $\dfrac{2\pi}{3}$: $\left(-\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right)$
• $135^\circ$ or $\dfrac{3\pi}{4}$: $\left(-\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)$
• $150^\circ$ or $\dfrac{5\pi}{6}$: $\left(-\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right)$
• $180^\circ$ or $\pi$: $(-1,0)$
• $210^\circ$ or $\dfrac{7\pi}{6}$: $\left(-\dfrac{\sqrt{3}}{2},-\dfrac{1}{2}\right)$
• $225^\circ$ or $\dfrac{5\pi}{4}$: $\left(-\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2}\right)$
• $240^\circ$ or $\dfrac{4\pi}{3}$: $\left(-\dfrac{1}{2},-\dfrac{\sqrt{3}}{2}\right)$
• $270^\circ$ or $\dfrac{3\pi}{2}$: $(0,-1)$
• $300^\circ$ or $\dfrac{5\pi}{3}$: $\left(\dfrac{1}{2},-\dfrac{\sqrt{3}}{2}\right)$
• $315^\circ$ or $\dfrac{7\pi}{4}$: $\left(\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2}\right)$
• $330^\circ$ or $\dfrac{11\pi}{6}$: $\left(\dfrac{\sqrt{3}}{2},-\dfrac{1}{2}\right)$
• $360^\circ$ or $2\pi$: $(1,0)$ (same as $0^\circ$)

From each pair $(\cos\theta,\sin\theta)$ you can obtain $\tan\theta$ by division.

Exact values and the Pythagorean identity

Exact values are tightly linked to the key identity
$$
\sin^2\theta + \cos^2\theta = 1.
$$

If you know one of $\sin\theta$ or $\cos\theta$ as an exact value for a special angle, you can find the other (up to sign) by solving
$$
\cos^2\theta = 1 - \sin^2\theta
\quad\text{or}\quad
\sin^2\theta = 1 - \cos^2\theta.
$$

For example, if $\theta = 30^\circ$, you can start from $\sin 30^\circ = \dfrac{1}{2}$ and compute
$$
\cos^2 30^\circ = 1 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4},
$$
so
$$
\cos 30^\circ = \pm\frac{\sqrt{3}}{2}.
$$
The quadrant tells you which sign is correct (for $30^\circ$, it is positive).

In later chapters on trigonometric identities and the unit circle, this identity is used frequently together with the exact values listed here.

Practice with exact values (no calculator)

When you work with exact values, you avoid decimal approximations and keep answers in root form, such as:

• $\dfrac{\sqrt{2}}{2}$ instead of $0.7071\ldots$
• $\dfrac{\sqrt{3}}{2}$ instead of $0.8660\ldots$
• $\dfrac{\sqrt{3}}{3}$ instead of $0.5773\ldots$

Being comfortable recognizing these root forms and knowing which angle they correspond to is the goal of this chapter. Subsequent trigonometry topics, like graphing trigonometric functions and using trigonometric identities, will rely heavily on these exact values.