Question

How do you find cos if \[\tan = \dfrac{3}{4}\]?

Answer 1

Hint:
Here, we will use the basic formula of the tan trigonometry functions to find out the value of two sides of the triangle. Then we will apply the Pythagoras theorem to find the hypotenuse of the triangle. Then we will simply apply the formula of the cos function to get its value.

Complete step by step solution:
Given is \[\tan = \dfrac{3}{4}\].
We know that the formula of the tan function is equal to
\[\tan = \] side opposite \[\div\] side adjacent
So on comparing the formula of the tan function with the given value of the tan function, we get
Side opposite \[= 3\]
Side Adjacent \[= 4\]
So, by using the Pythagoras theorem of a right triangle we will get the value of the hypotenuse of the triangle. Therefore, we get
\[ H = \sqrt {{3^2} + {4^2}} \]
Now by solving the above equation, we get
\[ \Rightarrow H = \sqrt {9 + 16} = \sqrt {25} = 5\]
Now we will find the value of the cos function using the formula of the cos function. Therefore, we get
\[ \Rightarrow \cos = \] Side adjacent \[\div\] H
So by putting the value of the adjacent side and the hypotenuse in the formula of the cos function. Therefore, we get
\[ \Rightarrow \cos = \dfrac{4}{5}\]

Hence the value of cos is equal to \[\dfrac{4}{5}\].

Additional information:
We know that the Pythagoras theorem is applied only to the right angled triangle. Right angled triangle is the triangle which has one of its angles equal to \[90{}^\circ \]. The hypotenuse is the longest side of the right angled triangle. We should know the basic condition of the Pythagoras theorem which states that the square of the hypotenuse is equal to the sum of the square of the other two sides of the triangle.

Note:
We should know the different properties of the trigonometric function and with the help of this concept this question can be easily solved. Trigonometry is the study of the relationship between the angles and sides of a right triangle. Right Triangle is a triangle where one of its interior angles is a right angle (90 degrees). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the hypotenuse. The sides adjacent to the right angle are called base.