Special values: π/6, π/4, π/3

Trigonometry, grade 10 (5 units) Lesson 4 of 21Learning

Special values: π/6, π/4, π/3

The angle

\displaystyle \frac{\pi}{4}

$\displaystyle \frac{\pi}{4}$ is halfway from $\displaystyle (1,0)$ to $\displaystyle (0,1)$ , so the point $\displaystyle P\left(\frac{\pi}{4}\right)$ is on the line $\displaystyle {y}={x}$ : its two coordinates are equal. Substitute $\displaystyle {x}={y}$ in the circle’s equation:

$\displaystyle {x^2+x^2}={1}$ , that is $\displaystyle {x^2}={\frac{1}{2}}$ ;
in the first quadrant $\displaystyle x>0$ , so $\displaystyle {x}={\frac{1}{\sqrt{2}}}={\frac{\sqrt{2}}{2}}$ .

\displaystyle \sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}

The angles

\displaystyle \frac{\pi}{6}

and

\displaystyle \frac{\pi}{3}

The points $\displaystyle P\left(\frac{\pi}{6}\right)$ and $\displaystyle P\left(-\frac{\pi}{6}\right)$ are symmetric about the $\displaystyle x$ -axis. With the origin $\displaystyle O$ they are the vertices of a triangle:

two of its sides are radii, of length 1;
the angle between them is $\displaystyle {\frac{\pi}{6}+\frac{\pi}{6}}={\frac{\pi}{3}}$ , that is $\displaystyle 60^{\circ}$ ;
so it is an equilateral triangle, and the third side – the upright segment between the two points – is of length 1 too.

The $\displaystyle x$ -axis cuts this segment in half, so $\displaystyle {\sin\frac{\pi}{6}}={\frac{1}{2}}$ . From the basic identity, $\displaystyle {\cos\frac{\pi}{6}}={\sqrt{1-\frac{1}{4}}}={\frac{\sqrt{3}}{2}}$ .

The point $\displaystyle P\left(\frac{\pi}{3}\right)$ is the reflection of $\displaystyle P\left(\frac{\pi}{6}\right)$ in the line $\displaystyle {y}={x}$ , so the two coordinates swap: $\displaystyle {\cos\frac{\pi}{3}}={\frac{1}{2}}$ , $\displaystyle {\sin\frac{\pi}{3}}={\frac{\sqrt{3}}{2}}$ .

The table of values

$\displaystyle t$	$\displaystyle 0$	$\displaystyle \frac{\pi}{6}$	$\displaystyle \frac{\pi}{4}$	$\displaystyle \frac{\pi}{3}$	$\displaystyle \frac{\pi}{2}$
$\displaystyle \sin t$	$\displaystyle 0$	$\displaystyle \frac{1}{2}$	$\displaystyle \frac{\sqrt{2}}{2}$	$\displaystyle \frac{\sqrt{3}}{2}$	$\displaystyle 1$
$\displaystyle \cos t$	$\displaystyle 1$	$\displaystyle \frac{\sqrt{3}}{2}$	$\displaystyle \frac{\sqrt{2}}{2}$	$\displaystyle \frac{1}{2}$	$\displaystyle 0$

Along the sine row the values grow, and the cosine row is the same row in reverse.

Reflections into the other quadrants

On the board: $\displaystyle {A}={P\left(\frac{\pi}{6}\right)}$ and its three reflections.

$\displaystyle {B}={P\left(\frac{5\pi}{6}\right)}$ – the reflection in the $\displaystyle y$ -axis: $\displaystyle {\frac{5\pi}{6}}={\pi-\frac{\pi}{6}}$ ;
$\displaystyle {C}={P\left(\frac{7\pi}{6}\right)}$ – the reflection in the origin: $\displaystyle {\frac{7\pi}{6}}={\pi+\frac{\pi}{6}}$ ;
$\displaystyle {D}={P\left(\frac{11\pi}{6}\right)}$ – the reflection in the $\displaystyle x$ -axis: $\displaystyle {\frac{11\pi}{6}}={2\pi-\frac{\pi}{6}}$ .

\displaystyle x^{2}+y^{2}=1

\displaystyle A=(\frac{\sqrt{3}}{2},\frac{1}{2})

\displaystyle B=(-\frac{\sqrt{3}}{2},\frac{1}{2})

\displaystyle C=(-\frac{\sqrt{3}}{2},-\frac{1}{2})

\displaystyle D=(\frac{\sqrt{3}}{2},-\frac{1}{2})

All four points have the same coordinates in absolute value, $\displaystyle \frac{\sqrt{3}}{2}$ and $\displaystyle \frac{1}{2}$ . Only the signs change, by the quadrant:

$\displaystyle {\sin\frac{5\pi}{6}}={\frac{1}{2}}$ , $\displaystyle {\cos\frac{5\pi}{6}}={-\frac{\sqrt{3}}{2}}$ ;
$\displaystyle {\sin\frac{7\pi}{6}}={-\frac{1}{2}}$ , $\displaystyle {\cos\frac{7\pi}{6}}={-\frac{\sqrt{3}}{2}}$ ;
$\displaystyle {\sin\frac{11\pi}{6}}={-\frac{1}{2}}$ , $\displaystyle {\cos\frac{11\pi}{6}}={\frac{\sqrt{3}}{2}}$ .

An exact value in two steps

To find an exact value of $\displaystyle \sin t$ or $\displaystyle \cos t$ :

write $\displaystyle t$ as $\displaystyle \pi-\alpha$ , $\displaystyle \pi+\alpha$ or $\displaystyle 2\pi-\alpha$ , where $\displaystyle \alpha$ is $\displaystyle \frac{\pi}{6}$ , $\displaystyle \frac{\pi}{4}$ or $\displaystyle \frac{\pi}{3}$ – the value in absolute value is the value at $\displaystyle \alpha$ ;
fix the sign by the quadrant of $\displaystyle P(t)$ .

For example, $\displaystyle {\frac{3\pi}{4}}={\pi-\frac{\pi}{4}}$ , and $\displaystyle P\left(\frac{3\pi}{4}\right)$ is in the second quadrant, so $\displaystyle {\cos\frac{3\pi}{4}}={-\frac{\sqrt{2}}{2}}$ .

Exercise 1

Find the exact value, without a calculator.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Work out the exact value.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

We look for all the numbers $\displaystyle t$ between $\displaystyle 0$ and $\displaystyle 2\pi$ for which $\displaystyle {\sin t}={\frac{1}{2}}$ .
These are the numbers whose point on the unit circle is at height $\displaystyle \frac{1}{2}$ .

On the board: the unit circle – its upper half $\displaystyle f$ and lower half $\displaystyle g$ – and the line $\displaystyle h$ , $\displaystyle {y}={\frac{1}{2}}$ .
Mark the points where the line meets the circle.

\displaystyle f(x)=\sqrt{1-x^{2}}

\displaystyle g(x)=-\sqrt{1-x^{2}}

\displaystyle h(x)=\frac{1}{2}

Which numbers $\displaystyle t$ between $\displaystyle 0$ and $\displaystyle 2\pi$ do these points belong to? Write them.

Formulas: trigonometry, grade 10 (5 units)

Open as a page

The unit circle (radians)

The unit circle
$\displaystyle x^2+y^2=1$
The point of the number $\displaystyle t$ on the unit circle
$\displaystyle P(t)=(\cos t,\ \sin t)$
All the numbers with the same point as $\displaystyle t$ ( $\displaystyle k$ an integer)
$\displaystyle t+2\pi k$
Radians and degrees
$\displaystyle \pi$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The length of an arc of $\displaystyle \alpha$ radians ( $\displaystyle R$ – the radius)
$\displaystyle \ell$ $\displaystyle =$ $\displaystyle \alpha R$
The area of a sector of $\displaystyle \alpha$ radians ( $\displaystyle R$ – the radius)
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}\alpha R^2$

Special values

$\displaystyle \sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}$
$\displaystyle \sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$
$\displaystyle \sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$
$\displaystyle \tan\frac{\pi}{6}=\frac{\sqrt{3}}{3},\quad\tan\frac{\pi}{4}=1,\quad\tan\frac{\pi}{3}=\sqrt{3}$

Identities

The basic identity
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =$ $\displaystyle 1$
Tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$ $\displaystyle \cos\alpha\neq0$
$\displaystyle 1+\tan^2\alpha$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2\alpha}$
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(\pi-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin(\pi+\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(\pi+\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \sin(\alpha+2\pi k)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \tan(\alpha+\pi k)$ $\displaystyle =$ $\displaystyle \tan\alpha$

Graphs of trigonometric functions

A general sine (and a cosine likewise)
$\displaystyle y=A\sin(ax+b)+B$
The period of $\displaystyle \sin(ax+b)$ and of $\displaystyle \cos(ax+b)$
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{2\pi}{|a|}$
The range of $\displaystyle A\sin(ax+b)+B$
$\displaystyle B-|A|\leq y\leq B+|A|$
The period of $\displaystyle \tan(ax+b)$
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{\pi}{|a|}$

Trigonometric equations ($k$ an integer)

The solutions of $\displaystyle {\sin x}={\sin\alpha}$
$\displaystyle x=\alpha+2\pi k,\quad x=\pi-\alpha+2\pi k$
The solutions of $\displaystyle {\cos x}={\cos\alpha}$
$\displaystyle x=\pm\alpha+2\pi k$
The solutions of $\displaystyle {\tan x}={\tan\alpha}$
$\displaystyle x=\alpha+\pi k$

Trigonometry in a triangle (degrees)

In a right-angled triangle: $\displaystyle a$ the leg opposite $\displaystyle \alpha$ , $\displaystyle b$ the leg beside it, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha=\frac{a}{c},\quad\cos\alpha=\frac{b}{c},\quad\tan\alpha=\frac{a}{b}$
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
The area of a triangle ( $\displaystyle \alpha$ – the angle between the sides $\displaystyle b$ and $\displaystyle c$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}bc\sin\alpha$
The area of a triangle ( $\displaystyle \beta$ and $\displaystyle \gamma$ – the angles at the side $\displaystyle a$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a^2\sin\beta\sin\gamma}{2\sin(\beta+\gamma)}$
The law of sines ( $\displaystyle R$ – the radius of the circle that circumscribes the triangle)
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines ( $\displaystyle \gamma$ – the angle between the sides $\displaystyle a$ and $\displaystyle b$ )
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$