Values at 30, 45 and 60 Degrees

Idea

Three angles keep coming back: 30, 45 and 60 degrees. You do not have to memorise their values. Two triangles hold all of them, and you can draw both.

Rule

Write the sides with a letter. The size is then free, and only the ratios survive.

30-60-90 is half an equilateral triangle of side $2x$ , so its sides are $x$ , $x\sqrt{3}$ and $2x$

45-45-90 is a square of side $x$ cut along its diagonal, so its sides are $x$ , $x$ and $x\sqrt{2}$

A right triangle

Try this. Compare the labeled lengths with sine, cosine and tangent. If size is adjustable, scale the triangle: lengths change together, but the ratios stay fixed.

θ = 30° · sin θ = 0.5 · cos θ = 0.866 · tan θ = 0.577

A right triangle

Try this. Compare the labeled lengths with sine, cosine and tangent. If size is adjustable, scale the triangle: lengths change together, but the ratios stay fixed.

θ = 45° · sin θ = 0.707 · cos θ = 0.707 · tan θ = 1

Every value in the table below is one side over another, and the $x$ always cancels.

How it is used

	$30^\circ$	$45^\circ$	$60^\circ$
$\sin$	$\dfrac{1}{2}$	$\dfrac{1}{\sqrt{2}}$	$\dfrac{\sqrt{3}}{2}$
$\cos$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{2}}$	$\dfrac{1}{2}$
$\tan$	$\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$
$\cot$	$\sqrt{3}$	$1$	$\dfrac{1}{\sqrt{3}}$
$\sec$	$\dfrac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$
$\csc$	$2$	$\sqrt{2}$	$\dfrac{2}{\sqrt{3}}$

The $30^\circ$ and $60^\circ$ columns are each other reversed, because those angles add to $90^\circ$ . The $45^\circ$ column is symmetric, because its triangle is.

Some books write $\dfrac{\sqrt{3}}{3}$ for $\tan 30^\circ$ and $\dfrac{\sqrt{2}}{2}$ for $\sin 45^\circ$ . Same numbers, root cleared from the bottom.

Degrees	$30^\circ$	$45^\circ$	$60^\circ$
Radians	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$

Calculus asks in radians, so learn the second row too.

Worked example

Evaluate $\sin\dfrac{\pi}{3} + \tan\dfrac{\pi}{4} \cdot \cos\dfrac{\pi}{6}$ .

Remember

$\pi = 180^\circ$ , half a turn degrees $\rightarrow$ radians: $\times \dfrac{\pi}{180}$ radians $\rightarrow$ degrees: $\times \dfrac{180}{\pi}$ $\dfrac{\pi}{3} \rightarrow \dfrac{\pi}{3} \cdot \dfrac{180}{\pi} = 60^\circ$

Convert, since the table is in degrees.

\sin 60^\circ + \tan 45^\circ \cdot \cos 30^\circ

Mark each value under the term it replaces.

\underbrace{\sin 60^\circ}_{\frac{\sqrt{3}}{2}} + \underbrace{\tan 45^\circ}_{1} \cdot \underbrace{\cos 30^\circ}_{\frac{\sqrt{3}}{2}}

Multiply before adding. Multiplying by $1$ changes nothing.

\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}

Two equal halves make the whole.

\sqrt{3}

Both $\sin 60^\circ$ and $\cos 30^\circ$ read the same side of the triangle below, once as the opposite and once as the adjacent. That is why they are equal.

A right triangle

Try this. Compare the labeled lengths with sine, cosine and tangent. If size is adjustable, scale the triangle: lengths change together, but the ratios stay fixed.

θ = 60° · sin θ = 0.866 · cos θ = 0.5 · tan θ = 1.732

Explore

The triangle below is the 30-60-90 triangle, with legs $1$ and $\sqrt{3}$ .

Pick sine and note which side lights up. Predict: if you switch to the other angle, what will sine read? Switch. The legs keep their lengths but trade names, so $\sin 60^\circ$ reads the same side as $\cos 30^\circ$ . That is why the two columns of the table are each other reversed.

Read the ratios

Try this. Choose a ratio to highlight its numerator and denominator. Switch the reference angle: opposite and adjacent exchange roles.

Read a ratioRead from

sin θ = opposite / hypotenuse = 1 / 2 = 0.5. Blue: numerator. Green: denominator.

Values at 30, 45 and 60 Degrees

Idea

Rule

How it is used

Worked example

Try it yourself.

Explore

Pause before the next idea.