The cosine ratio is one of the 3 main trigonometric ratios found in a right triangle.
In a right triangle, the cosine of an angle |(\boldsymbol \theta)| is the ratio between the length of the leg that is adjacent to the angle and the hypotenuse.||\cos \theta=\dfrac{\text{leg $\color{#333fb1}{\text{adjacent }}$to angle }\theta}{\text{$\color{#3A9A38}{\text{hypotenuse}}$}}||
If we want to determine the cosine of the acute angles in the following right triangle, we get 2 ratios.
The cosine trigonometric ratio is only used with respect to the acute angles of a right triangle. The cosine is never sought in relation to the right angle.
To find the length of a leg of a right triangle using the cosine ratio, the length of the hypotenuse and the measure of the angle adjacent to the side in question must be known.
Find the length of side |\overline{BC}| using the cosine ratio in the following right triangle.
Choose which angle to use to determine the length of |\overline{BC}| directly using the cosine ratio.
Since side |\overline{BC}| is opposite the |34^{\circ}| angle and not adjacent to it, the cosine ratio cannot be used with respect to this angle.
Side |\overline{BC}| is adjacent to the |56^{\circ}| angle. Since the length of the hypotenuse is also known, we can use the cosine ratio with respect to the angle |\theta| of |56^{\circ}| to find the measure of side |\overline{BC}.|
Substitute the measurements in the ratio.||\begin{align}\cos \theta&=\dfrac{\text{leg $\color{#333fb1}{\text{adjacent}}$ to angle }\theta}{\text{$\color{#3A9A38}{\text{hypotenuse}}$}}\\\cos56^{\circ}&=\dfrac{\color{#333fb1}{\text{m}\overline{BC}}}{\color{#3A9A38}{7.2}}\end{align}||Using a calculator, calculate the cosine of |56^{\circ}| and isolate the desired measurement.||\begin{align}0.5592&\approx\dfrac{\color{#333fb1}{\text{m}\overline{BC}}}{\color{#3A9A38}{7.2}}\\4.0262&\approx\color{#333fb1}{\text{m}\overline{BC}}\end{align}||
Answer: Rounded to the nearest hundredth, side |\overline{BC}| measures approximately |4.03\ \text{cm}.|
For greater accuracy, do the calculations in one step on a calculator. If this is not possible, keep at least 3 to 4 digits after the decimal point.
Here is an example when |\theta=65^{\circ}| and the hypotenuse measure |59\ \text{cm}.|
Performing the calculation in one step
||\begin{align}\cos65^{\circ}&=\dfrac{a}{{59}}\\\cos65^{\circ}\times59&=a\\\color{#EC0000}{24.93}&\approx a\end{align}||
Performing the calculation in 2 steps
||\begin{align}\cos65^{\circ}&=\dfrac{a}{59}\\\color{#ec0000}{0.42}&\approx\dfrac{a}{59}\\0.42\times59&\approx a\\\color{#EC0000}{24.78}&\approx a\end{align}||
If the calculations are performed in 2 steps, and we only round to 2 decimal places when calculating the cosine of the angle, there will be a difference of 15 hundredths compared to the correct answer.
In some cases, neither the length of the hypotenuse nor the length of the side adjacent to the angle are known. In order to apply the cosine ratio, the measure of the other angle has to be determined.
Use the cosine ratio to determine the length of the leg sought in the following right triangle.
Choose the angle to use to determine the cosine ratio in order to find the side measure sought.
Since side |\overline{BC}| is opposite the |69^{\circ}| angle and not adjacent to it, the cosine ratio cannot be used with respect to this angle.
Therefore, the measure of the other angle must be found.||\begin{align}180^{\circ}-90^{\circ}-69^{\circ}&=\text{m}\angle ABC\\21^{\circ}&=\text{m}\angle ABC\end{align}||
Side |\overline{BC}| is adjacent to the |21^{\circ}| angle. Since the length of the hypotenuse is also known, the cosine ratio can be used with respect to angle |\theta| of |21^{\circ}.|||\begin{align}\cos \theta&=\dfrac{\text{leg $\color{#333fb1}{\text{adjacent}}$ to angle }\theta}{\text{$\color{#3A9A38}{\text{hypotenuse}}$}}\\\cos21^{\circ}&=\dfrac{\color{#333fb1}{\text{m}\overline{BC}}}{\color{#3A9A38}{34.5}}\\\cos21^{\circ}\times \color{#3A9A38}{34.5}&=\color{#333fb1}{\text{m}\overline{BC}}\\32.21&\approx\color{#333fb1}{\text{m}\overline{BC}}\end{align}||
Answer: The leg sought measures approximately |32.21\ \text{cm}.|
In the previous example, the sine trigonometric ratio could have been used to find the length of side |\overline{BC}| using the |69^{\circ}| angle. The answer would have been the same.
To find the length of the hypotenuse in a right triangle using the cosine ratio, the measure of one acute angle and the length of the side adjacent to it must be known.
Find the length of the hypotenuse using the cosine ratio in the following right triangle.
Side |\overline{BC}| is adjacent to the |43^{\circ}| angle. Since we want to find the length of the hypotenuse, we can use the cosine ratio with an angle |\theta| of |43^{\circ}.|
Substitute the measures in the ratio.||\begin{align}\cos \theta&=\dfrac{\text{leg $\color{#333fb1}{\text{adjacent}}$ to angle }\theta}{\text{$\color{#3A9A38}{\text{hypotenuse}}$}}\\\cos43^{\circ}&=\dfrac{\color{#333fb1}{4.52}}{\color{#3A9A38}{\text{m}\overline{AC}}}\end{align}||Since the length of the hypotenuse is found in the denominator of the ratio, proceed as follows.||\begin{align}\dfrac{\cos43^{\circ}}{1}&=\dfrac{4.52}{\color{#3A9A38}{\text{m}\overline{AC}}}\\\color{#3A9A38}{\text{m}\overline{AC}}\times \cos43^{\circ}&=4.52\times 1\\\color{#3A9A38}{\text{m}\overline{AC}}&=\dfrac{4.52}{\cos43^{\circ}}\\\color{#3A9A38}{\text{m}\overline{AC}}&\approx 6.18\end{align}||
Answer: The hypotenuse measures approximately |6.18\ \text{cm}.|
In a right triangle, the length of the leg adjacent to a |60^{\circ}| angle is always equal to half the length of the hypotenuse.
This means the cosine ratio is always |\dfrac{1}{2}.|||\begin{align}\cos60^{\circ}&=\dfrac{\text{leg $\color{#333fb1}{\text{adjacent}}$ to angle }\theta}{\text{$\color{#3A9A38}{\text{hypotenuse}}$}}\\&=\dfrac{\frac{\color{#3A9A38}c}{2}}{\color{#3A9A38}c}\\&=\dfrac{1}{2}\end{align}||
What is the length of side |\overline{AB}| in the following triangle?
Since the right triangle has a |60^{\circ}| angle, we know the length of the side that is adjacent to it is half the length of the hypotenuse. So, the hypotenuse is 2 times |4.5.|||\overline{AB}=4.5\times 2=9\ \text{cm}||Answer: Side |\overline{AB}| measures |9\ \text{cm}.|
To find the measure of an acute angle in a right triangle using the cosine ratio, the length of the side adjacent to it and the length of the hypotenuse must be known. This is essentially the same as answering the following question: “What angle gives a cosine of...?”
First, determine the cosine ratio, then use the |\cos^{-1}| key (also called |arccos| ) on a calculator.
Find the measure of angle |BAC| in the following right triangle using the cosine ratio.
The only information we have is the length of the side opposite angle |BAC| and the length of the side adjacent to it. This means the cosine ratio cannot be applied. The length of the hypotenuse must be determined first. Since the 2 legs of the right triangle are known, the Pythagorean theorem can be applied to find its length.||\begin{align}\left(\text{m}\overline{AC}\right)^2+\left(\text{m}\overline{BC}\right)^2&=\left(\text{m}\overline{AB}\right)^2\\9.65^2+3.9^2&=\left(\text{m}\overline{AB}\right)^2\\\color{#EC0000}{\sqrt{\color{black}{108.3325}}}&=\color{#EC0000}{\sqrt{\color{black}{\left(\text{m}\overline{AB}\right)^2}}}\\10.41&\approx \text{m}\overline{AB}\end{align}||The hypotenuse measures approximately |10.41\ \text{cm}.| The cosine ratio can now be applied to find the measure of angle |BAC.|||\begin{align}\cos \theta &=\dfrac{\text{leg $\color{#333fb1}{\text{adjacent}}$ to angle }\theta}{\text{$\color{#3A9A38}{\text{hypotenuse}}$}}\\\cos \theta&=\dfrac{\color{#333fb1}{9.65}}{\color{#3a9a38}{10.41}}\\\theta&=\color{#EC0000}{\cos^{-1}\left(\color{black}{\dfrac{9.65}{10.41}}\right)}\\\theta&\approx22^{\circ}\end{align}||Answer: Angle |BAC| measures approximately |22^{\circ}.|
In trigonometry, there are several possible ways to get the same answer. In the previous example, we could have used |\boldsymbol{\tan^{-1}}| with respect to angle |BAC,| which would have given the measurement directly.
Here is part of a trigonometric table, which is a tool that was used in mathematics before the invention of the calculator. It includes the measures of certain angles and their corresponding cosine ratios.
At the time, this table had to be used to find the angle measure in a right triangle. For example, if it was calculated that the cosine ratio was approximately |0.6428,| then the angle sought would measure |50^{\circ}.|
| Angle measure | Cosine ratio |
|---|---|
| |10^{\circ}| | |\approx 0.9848| |
| |20^{\circ}| | |\approx 0.9397| |
| |30^{\circ}| | |\approx 0.866| |
| |40^{\circ}| | |\approx 0.766| |
| |50^{\circ}| | |\approx 0.6428| |
| |60^{\circ}| | |0.5| |
Thankfully, since calculators were invented, we no longer need to use the trigonometric table to determine the measure of an angle in a right triangle.
The arccosine function (denoted |\cos^{-1}(x)| ) is the inverse of the cosine function.||\cos \theta=x\ \Leftrightarrow \ \cos^{-1}x=\theta||