The sine, cosine and tangent of an acute angle

Geometry, grade 10 (4 units) Lesson 9 of 20Learning

The sine, cosine and tangent of an acute angle

Three ratios in a right-angled triangle

In a right-angled triangle, each acute angle $\displaystyle \alpha$ has the side opposite the angle, the side adjacent to the angle, and the hypotenuse – the side opposite the right angle. The three ratios between them are the trigonometric functions of $\displaystyle \alpha$ :

sine – the opposite side over the hypotenuse:

\displaystyle \sin\alpha

\displaystyle =

\displaystyle \frac{a}{c}

cosine – the adjacent side over the hypotenuse:

\displaystyle \cos\alpha

\displaystyle =

\displaystyle \frac{b}{c}

tangent – the opposite side over the adjacent side:

\displaystyle \tan\alpha

\displaystyle =

\displaystyle \frac{a}{b}

In the drawing, for $\displaystyle \alpha$ : $\displaystyle a$ is the side opposite it, $\displaystyle b$ the side adjacent to it, and $\displaystyle c$ the hypotenuse. In this course angles are measured in degrees, and the calculator is set to degrees (DEG).

A ratio that depends only on the angle

Two right-angled triangles with the same acute angle $\displaystyle \alpha$ have two equal angles: the right angle and $\displaystyle \alpha$ . So they are similar (AA), and their corresponding sides are in the same ratio:

\frac{BC}{AB}=\frac{EF}{AE}

In the drawing, $\displaystyle ABC$ and $\displaystyle AEF$ : the sine of $\displaystyle \alpha$ comes out the same number in both, and so do the cosine and the tangent. The values depend only on the size of the angle, not on the size of the triangle – so a table or a calculator can give them.

Exercise 1

In the right-angled triangle in the drawing, the legs are $\displaystyle 8$ and $\displaystyle 15$ , and the hypotenuse $\displaystyle 17$ .
Write the ratio as a fraction in its lowest terms.

\displaystyle \sin\alpha

/ – 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

\displaystyle \cos\alpha

/ – 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

\displaystyle \tan\alpha

/ – 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

\displaystyle \tan B

/ – 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

The values as the angle grows

The hypotenuse is longer than both legs, so the sine and the cosine of an acute angle are between $\displaystyle 0$ and $\displaystyle 1$ . When an acute angle grows from $\displaystyle 0^{\circ}$ to $\displaystyle 90^{\circ}$ , with the hypotenuse kept the same length:

the side opposite it grows longer – the sine increases, from $\displaystyle 0$ to $\displaystyle 1$ ;
the side adjacent to it grows shorter – the cosine decreases, from $\displaystyle 1$ to $\displaystyle 0$ ;
the tangent increases from $\displaystyle 0$ , without a limit: the adjacent side is close to $\displaystyle 0$ , and dividing by it gives very large numbers.

Exercise 2

Calculate with a calculator, and round to two decimal places (the last line starts with $\displaystyle \approx$ ).

/ – 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

Is $\displaystyle \sin40^{\circ}$ twice $\displaystyle \sin20^{\circ}$ ?
And is $\displaystyle \sin80^{\circ}$ twice $\displaystyle \sin40^{\circ}$ ?
What does this say about $\displaystyle \sin(2\alpha)$ and $\displaystyle 2\sin\alpha$ ?

The angles 0° and 90°

When the angle is very close to $\displaystyle 0^{\circ}$ , the triangle is flat: the side opposite it is almost $\displaystyle 0$ , and the side adjacent to it almost as long as the hypotenuse. When it is close to $\displaystyle 90^{\circ}$ – the other way round. So we define:

$\displaystyle {\sin0^{\circ}}={0}$ , $\displaystyle {\cos0^{\circ}}={1}$ , $\displaystyle {\tan0^{\circ}}={0}$ ;
$\displaystyle {\sin90^{\circ}}={1}$ , $\displaystyle {\cos90^{\circ}}={0}$ ;
$\displaystyle \tan90^{\circ}$ is not defined: a division by $\displaystyle 0$ .

An angle from its sine

Between $\displaystyle 0^{\circ}$ and $\displaystyle 90^{\circ}$ the sine increases, so each of its values belongs to only one acute angle. The calculator finds it with the $\displaystyle \sin^{-1}$ key (and likewise $\displaystyle \cos^{-1}$ , $\displaystyle \tan^{-1}$ ):

$\displaystyle {\sin\alpha}={0.5}$ , so $\displaystyle {\alpha}={\sin^{-1}0.5}={30^{\circ}}$ ;
$\displaystyle {\tan\alpha}={\frac{3}{4}}$ , so $\displaystyle {\alpha}={\tan^{-1}\frac{3}{4}\approx36.87^{\circ}}$ .

Exercise 3

Find the angle, and round to two decimal places.

/ – 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 4

Is there an acute angle whose sine is $\displaystyle 1.2$ ?
And an acute angle whose tangent is $\displaystyle 1.2$ ?
Explain.

Formulas: geometry, grade 10 (4 units)

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Analytic geometry

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$
The midpoint of the segment
$\displaystyle M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$
The slope of a line through two points
$\displaystyle m$ $\displaystyle =$ $\displaystyle \frac{y_2-y_1}{x_2-x_1}$ $\displaystyle x_1\neq x_2$
A line’s equation in explicit form – $\displaystyle m$ the slope, $\displaystyle b$ where it meets the $\displaystyle y$ -axis
$\displaystyle y=mx+b$
A line’s equation in implicit form
$\displaystyle Ax+By+C=0$
The line of slope $\displaystyle m$ through the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
Parallel lines
$\displaystyle m_1$ $\displaystyle =$ $\displaystyle m_2$
Perpendicular lines
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$
The slope – $\displaystyle \alpha$ the angle between the line and the positive direction of the $\displaystyle x$ -axis
$\displaystyle m$ $\displaystyle =$ $\displaystyle \tan\alpha$

Similar triangles and midsegments

The similarity ratio – corresponding sides
$\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k$
The ratio of the perimeters
$\displaystyle \frac{P_1}{P_2}$ $\displaystyle =$ $\displaystyle k$
The ratio of the areas
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$
The ratio of corresponding altitudes, medians and angle bisectors
$\displaystyle \frac{h_1}{h_2}=\frac{m_1}{m_2}=\frac{d_1}{d_2}=k$
A triangle’s midsegment – $\displaystyle DE$ parallel to $\displaystyle BC$
$\displaystyle DE$ $\displaystyle =$ $\displaystyle \frac{1}{2}BC$
A trapezoid’s midsegment – $\displaystyle a$ , $\displaystyle b$ the bases
$\displaystyle EF$ $\displaystyle =$ $\displaystyle \frac{a+b}{2}$

Trigonometry

Sine – $\displaystyle a$ the side opposite the angle, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
Cosine – $\displaystyle b$ the side adjacent to the angle
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
Tangent – the opposite side over the adjacent side
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The tangent from the sine and the cosine
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$
The identity from Pythagoras
$\displaystyle \sin^{2}\alpha+\cos^{2}\alpha$ $\displaystyle =$ $\displaystyle 1$
Angles whose sum is $\displaystyle 90^{\circ}$
$\displaystyle \sin(90^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos(90^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The sine of an obtuse angle
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
A triangle’s area – $\displaystyle \gamma$ the angle between the sides $\displaystyle a$ and $\displaystyle b$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$

Exact values

$\displaystyle \sin30^{\circ}=\cos60^{\circ}=\frac{1}{2}$
$\displaystyle \sin45^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}$
$\displaystyle \sin60^{\circ}=\cos30^{\circ}=\frac{\sqrt{3}}{2}$
$\displaystyle \tan30^{\circ}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{3}}{3}$
$\displaystyle \tan45^{\circ}$ $\displaystyle =$ $\displaystyle 1$
$\displaystyle \tan60^{\circ}$ $\displaystyle =$ $\displaystyle \sqrt{3}$

Special lines in a triangle

Where the medians meet, $\displaystyle G$ divides the median $\displaystyle AD$ in the ratio $\displaystyle 2:1$
$\displaystyle AG$ $\displaystyle =$ $\displaystyle 2GD$
Where the medians meet, in coordinates
$\displaystyle G\left(\frac{x_1+x_2+x_3}{3},\ \frac{y_1+y_2+y_3}{3}\right)$
The median $\displaystyle AD$ splits the triangle into two triangles of equal area
$\displaystyle S_{ABD}$ $\displaystyle =$ $\displaystyle S_{ACD}$
The angle bisector theorem – $\displaystyle AD$ bisects the angle $\displaystyle A$
$\displaystyle \frac{BD}{DC}$ $\displaystyle =$ $\displaystyle \frac{AB}{AC}$