Sine, Cosine and Tangent

The following relationships result from this:

• $\sin(\alpha)=\frac{a}{c}$

• $\cos(\alpha)=\frac{b}{c}$

• $\tan(\alpha)=\frac{a}{b}$

Important function values

The following table of values shows the function values of the cosine, sine and tangent:

Attention: In the case $\alpha=90°$ we cannot form a triangle, because $\tan\left(90°\right)$ is not defined.

Dependencies

If you are given a right triangle, one side and one angle, you can use the trigonometric functions to calculate the remaining side lengths.

You obtain these formulas by resolving the definitions of sine, cosine and tangent for $b$, $a$ and $c$ respectively.

In the first case, if the hypotenuse $c$ is given, it works as follows.

$\sin\alpha=\dfrac a c \Rightarrow a=\sin\alpha \cdot c$

$\cos\alpha=\dfrac b c \Rightarrow b=\cos \alpha\cdot c$

The other cases work in the same way.

Example

For a right triangle $\bigtriangleup\mathrm{ABC}$ at $C$, the length of the hypotenuse $c=4$ and the angle $\alpha=30^\circ$ are given(first diagram).

Then the lengths of the adjacent side $b$ and the opposite side $a$ can be calculated using the sine and the cosine:

$b\;=\;\cos(\alpha)\cdot c=\;\cos(30^\circ)\cdot4\;=\;\dfrac{\sqrt3}2\cdot4\;=\;2\sqrt3$

$a\;=\;\sin(\alpha)\cdot c\;=\;\sin(30^\circ)\cdot4\;=\;\dfrac12\cdot4\;=\;2$

Calculation rules

There are some calculation rules for sine, cosine and tangent. Here, you learn about

• writing the tangent as quotient of sine and cosine,

• the trigonometric Pythagoras,

• the addiotion theorems.

Tangent as quotient of sine and cosine

Directly via the definition from above, one obtains the following alternative representation for the tangent:

You can also simply recalculate the correctness of this equation:

$\dfrac {\sin\alpha}{\cos\alpha}= \dfrac {\dfrac a c} {\dfrac b c}= \dfrac {a\cdot c}{b \cdot c}= \dfrac a b = \tan\alpha$

Trigonometric Pythagoras

From the definition on the unit circle it follows directly from the Pythagorean theorem:

You can find a detailed explanation in the video below.

Addition theorems

The addition theorems make it possible to calculate the sine and cosine of a sum:

Further relationships between sine, cosine and tangent

In the article relations between trigonometric functions you can find more relationships of the functions.

Trigonometry on the unit circle

You can also use the unit circle to illustrate the angle relationships shown in the article. You can read more about this topic here: Trigonometry on the unit circle.

Sine, cosine and tangent function

You can also represent sine, cosine and tangent as functions. You can find out more about this in the article sine function and cosine function or tangent function.