In right-angled triangles, the trigonometric functions can be used to construct simple relationships between the sides and angles of the triangle. These can be referred to as the trigonometric ratios.
Notice that if we instead choose \angle BCA to be \theta, the opposite and adjacent sides will switch to match the angle's new position.
Using the trigonometric functions and the given angle, we can express the ratios between the different pairs of sides as:\sin \theta =\dfrac{\text{Opposite }}{\text{Hypotenuse }} \quad \quad \cos \theta =\dfrac{\text{Adjacent }}{\text{Hypotenuse }} \quad \quad \tan \theta =\dfrac{\text{Opposite }}{\text{Adjacent }}
If we are given one of the angles in a right-angled triangle and all of the side lengths, we can write the trigonometric ratios as fractions of the side lengths.
If we are only given two of the side lengths, we can calculate the third using Pythagoras' theorem.
For example:
We can then identify all the sides by considering their position with respect to the angle \theta.
\text{Hypotenuse}= 17
\text{Opposite}= 15
\text{Adjacent}= 8
Using these values, we can then find the trigonometric ratios:
\sin \theta =\dfrac{\text{Opposite }}{\text{Hypotenuse }}= \dfrac{15}{17} \quad \quad \cos \theta =\dfrac{\text{Adjacent }}{\text{Hypotenuse }}= \dfrac{8}{17} \quad \quad \tan \theta =\dfrac{\text{Opposite }}{\text{Adjacent }}= \dfrac{15}{8}
As long as we are given one angle (that is not the right angle) and at least two side lengths, we can find the trigonometric ratios for any right-angled triangle.
The applet below uses the mnemonic SOHCAHTOA in finding the trigonometric ratios. Move the slider to adjust the value of angle \theta.
For angle 0 \leq \theta \leq 90\degree: 0 \leq \sin \theta \leq 1, 0 \leq \cos \theta \leq 1, and \tan \theta \geq 0.
Evaluate \sin \theta within \triangle ABC.
Find the value of \tan \theta in \triangle ABC.
If we are only given two of the side lengths, we can calculate the third using Pythagoras' theorem: c^2=a^2+b^2.