Math Trigonometry Sine, Cosine and Tangent

# Sine, Cosine and Tangent

With Sine, Cosine and Tangent in each right-angled triangle, you can calculate the angle of the adjacent/opposite, or the angle itself, if two of the three quantities are known.

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### Remember

• Sine: $\sin=\frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine: $\cos=\frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent: $\tan=\frac{\text{opposite}}{\text{adjacent}}$

If you use the pages now, we find in our triangle ABC with $\gamma=90^\circ$:

• $\sin(\alpha)=\frac{a}{c}$ and $\sin(\beta)=\frac{b}{c}$
• $\cos(\alpha)=\frac{b}{c}$ and $\cos(\beta)=\frac{a}{c}$
• $\tan(\alpha)=\frac{a}{b}$ and $\tan(\beta)=\frac{b}{a}$

### Example

Given is a right triangle with $\beta=90^\circ$, $\alpha=60^\circ$ and $c=4$. Calculate $b$.

1. #### Find the right formula

$\cos=\frac{\text{adjacent}}{\text{hypotenuse}}$
$\cos(\alpha)=\frac{c}{b}$
2. #### Change the formula

$\cos(\alpha)=\frac{c}{b}\quad|\cdot b$
$\cos(\alpha)\cdot b=c\quad|:\cos(\alpha)$
$b=\frac{c}{\cos(\alpha)}$
3. #### Insert and calculate

$b=\frac{4}{\cos(60^\circ)}=8$