5.4 Exact trigonometric values — Learn

Prerequisites: Pythagoras' theorem; simplifying surds; the four quadrants of the coordinate plane; the graphs of sine, cosine and tangent.

By the end of this section you can: derive the exact values of sine, cosine and tangent at $30^\circ$, $45^\circ$ and $60^\circ$; use them at any angle with the correct sign; convert between degrees and radians for these angles.

1. Two special triangles

Two right-angled triangles give every exact value in this section.

The half-square. A square of side $1$ cut along a diagonal gives a right-angled triangle with two $45^\circ$ angles. Its hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$. Therefore

The half-equilateral triangle. An equilateral triangle of side $2$ cut in half gives a right-angled triangle with angles $30^\circ$ and $60^\circ$. Its sides are $1$, $\sqrt{3}$ and $2$. Therefore

2. The table to remember

The sine row increases from $0$ to $1$. The cosine row decreases from $1$ to $0$. That pattern is the quickest way to remember which surd belongs where.

3. Signs in the four quadrants

The exact values above are all positive. For angles outside the first quadrant, find the related acute angle with the horizontal axis, then attach the sign.

A common way to record this is the word CAST, read anticlockwise from the fourth quadrant: cosine, all, sine, tangent.

Example 1. Find the exact value of $\sin\dfrac{5\pi}{6}$.

$\dfrac{5\pi}{6}$ is $150^\circ$, in the second quadrant, where sine is positive. The related acute angle is $\pi - \dfrac{5\pi}{6} = \dfrac{\pi}{6}$. So

Example 2. Find the exact value of $\cos\dfrac{4\pi}{3}$.

$\dfrac{4\pi}{3}$ is $240^\circ$, in the third quadrant, where cosine is negative. The related acute angle is $\dfrac{4\pi}{3} - \pi = \dfrac{\pi}{3}$. So

Example 3. Find the exact value of $\tan\dfrac{7\pi}{4}$.

$\dfrac{7\pi}{4}$ is $315^\circ$, in the fourth quadrant, where tangent is negative. The related acute angle is $2\pi - \dfrac{7\pi}{4} = \dfrac{\pi}{4}$. So

Watch out. Three mistakes are common with exact values.

Swapping sine and cosine. At $30^\circ$, $\sin 30^\circ = \dfrac{1}{2}$ and $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$. The larger value goes with the smaller angle for sine. Check against the graph: sine starts near $0$ and cosine starts near $1$.

Losing the minus sign. Find the quadrant first, then attach the sign. An answer such as $\cos 150^\circ = \dfrac{\sqrt{3}}{2}$ is missing a minus, because $150^\circ$ lies in the second quadrant and cosine is negative there.

Writing a value for $\tan 90^\circ$. At $90^\circ$ the cosine is $0$, and division by zero is undefined. The tangent has an asymptote there. Re-check: for each answer, ask which quadrant the angle is in, and whether the function is positive there.

You should now be able to

• derive the exact values from the half-square and the half-equilateral triangle;
• state the exact value of sine, cosine and tangent at $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$;
• use the same values in radians;
• find the related acute angle and attach the correct sign for any quadrant;
• explain why $\tan 90^\circ$ is undefined.