Question

If the value of $\sec \theta =\dfrac{25}{7}$, find all trigonometric ratios of $\theta $.

Answer 1

Hint: We are given $\sec \theta =\dfrac{25}{7}$, take the reciprocal we will get $\cos \theta $ and then from $\cos \theta $ we will get $\sin \theta $. After that use the formula of $\sec \theta $ and $\tan \theta $, you will get $\tan \theta $ and from that taking reciprocal will give $\cot \theta $.

Complete step-by-step answer:
We are given $\sec \theta =\dfrac{25}{7}$.
Now we know that $\cos \theta =\dfrac{1}{\sec \theta }$.
So, we get, $\cos \theta =\dfrac{7}{25}$.
Now we know that, $\sin \theta =\sqrt{1-{{\cos }^{2}}\theta }$.
$\sin \theta =\sqrt{1-{{\left( \dfrac{7}{25} \right)}^{2}}}$
Simplifying we get,
$\sin \theta =\sqrt{\dfrac{625-49}{625}}$
$\sin \theta =\sqrt{\dfrac{576}{625}}$
$\sin \theta =\dfrac{24}{25}$
Also, $\text{cosec}\theta =\dfrac{1}{\sin \theta }$
We get,
$\text{cosec}\theta =\dfrac{25}{24}$
Now we have $\sin \theta $ and $\cos \theta $.
Here we know, $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$.
Taking values from above we get,
$\tan \theta =\dfrac{\dfrac{24}{25}}{\dfrac{7}{25}}=\dfrac{24}{7}$
We get, $\tan \theta =\dfrac{24}{7}$.
Now reciprocal of $\tan \theta $ is $\cot \theta $.
So $\cot \theta =\dfrac{7}{24}$
Therefore, we have got all the trigonometric ratios.
So, we have,
$\sin \theta =\dfrac{24}{25}$
$\cos \theta =\dfrac{7}{25}$
$\sec \theta =\dfrac{25}{7}$
$\text{cosec}\theta =\dfrac{25}{24}$
$\tan \theta =\dfrac{24}{7}$
$\cot \theta =\dfrac{7}{24}$

Additional information:
Trigonometry is one of the most important branches of mathematics. In trigonometry, trigonometric ratios are derived from the sides of a right-angled triangle. There are six 6 ratios such as sine, cosine, tangent, cotangent, cosecant, and secant. There are many trigonometry formulas and trigonometric identities, which are used to solve complex equations in geometry. It is defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle. The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle. Trigonometric functions are also known as a Circular Functions can be simply defined as the functions of an angle of a triangle. It means that the relationship between the angles and sides of a triangle are given by these trig functions.

Note: There are a number of trigonometric formulas and identities which denotes the relation between the functions and help to find the angles of the triangle. Each trigonometric function is interrelated to each other.