Question

How do you find the exact values of \[\cos \theta \] and \[\sin \theta \] when\[\tan \theta =\dfrac{5}{12}\]?

Answer 1

Hint: The given question is the trigonometric expression and in order to solve this solve we have to use the properties of trigonometric functions. First we need to get the relation of sine, cosine and the tangent function. We have given the value of tan function, then by using Pythagoras theorem we will find the value of hypotenuse. Later putting the value for each trigonometric function, we will get the required exact values.

Formula used:
To find the tangent of any angle we need to just divide the sine and cosine of the same angle:\[\tan \theta =\dfrac{\sin \theta }{\cos \theta }=\dfrac{perpendicular}{base}\]

Complete step by step solution:
We have given that,
\[\tan \theta =\dfrac{5}{12}\]
As we know that,
The tangent of any angle we need to just divide the sine and cosine of the same angle:\[\tan \theta =\dfrac{\sin \theta }{\cos \theta }=\dfrac{perpendicular}{base}\]

We know that:
\[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\], where
\[\sin \theta =\dfrac{perpendicular}{hypotenuse}\]
\[\cos \theta =\dfrac{base}{hypotenuse}\]
Now if we divide \[\sin \theta \] by \[\cos \theta \],, we will get
\[\dfrac{\sin \theta }{\cos \theta }=\dfrac{\dfrac{perpendicular}{hypotenuse}}{\dfrac{base}{hypotenuse}}=\dfrac{perndicular}{base}=\tan \theta \]\[\]
Hence we get the relation between sin, cos and tan.
Therefore,
We have given the following function, i.e.
\[\tan \theta =\dfrac{5}{12}\]
Thus,
\[\tan \theta =\dfrac{5}{12}=\dfrac{perpendicular\ side}{base\ side}\]
By using the Pythagoras theorem;
\[hypotenuse=\sqrt{{{\left( base\ side \right)}^{2}}+{{\left( perpendicular\ side \right)}^{2}}}\]
Therefore,
Hypotenuse = \[\sqrt{{{5}^{2}}+{{12}^{2}}}=\sqrt{25+144}=\sqrt{169}\]
The \[\sqrt{169}\] is equal to 13,
Thus hypotenuse = 13.
Now, putting the value.
Therefore, we have
\[\Rightarrow \sin \theta =\dfrac{perpendicular}{hypotenuse}=\dfrac{5}{13}\]
\[\Rightarrow \cos \theta =\dfrac{base}{hypotenuse}=\dfrac{12}{13}\]
Hence, it is the required possible answer.

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometric and the trigonometric ratios as well. It will make questions easier to solve. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.