Question

If $\sin \theta {\text{ = }}\dfrac{3}{5}$ and $\theta$ is acute, how do you calculate $\cos \theta$ and $\tan \theta$ ?

Answer 1

As we all know that in a right angle triangle $\sin \theta$ is the ratio of perpendicular to hypotenuse so by the ratio given we can complete the triangle using Pythagoras theorem and then we will be able to evaluate cos and tan for the triangle.

Complete step-by-step solution:
As given, $\sin \theta {\text{ = }}\dfrac{3}{5}$ and this is equal to $\dfrac{{perpendicular}}{{hypotenuse}}$
So as per the Pythagoras theorem,
$\Rightarrow bas{e^2} = hypotenuse{e^2} - perpendicular{r^2}$
Putting in the value of hypotenuse and perpendicular, we will get,
$\Rightarrow bas{e^2} = {5^2} - {3^2}$
Solving the above equation will give us,
$\Rightarrow bas{e^2} = 16$
Now taking under root will give us,
$\Rightarrow base = 4$
Now as we have the ratio of all three side of triangle we can calculate cos and tan,
$\Rightarrow \cos \theta = \dfrac{{base}}{{hypotenuse}}$
Putting in the value of base and hypotenuse will give us,
$\Rightarrow \cos \theta = \dfrac{4}{5}$
Similarly
$\Rightarrow \tan \theta = \dfrac{{perpendicular}}{{base}}$
Again, putting in the value of perpendicular and base will give us,
$\Rightarrow \tan \theta = \dfrac{3}{4}$

**Additional Information: **Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the ${90^ \circ }$ angle.

Note: The above question can also be solved with the help of trigonometric identities as well, if you remember that and value of sin is provided here so you can calculate cos and then the ratio of sin and cos is tan.