 Questions & Answers    Question Answers

# If $\sin \theta =\dfrac{3}{5}$, find the values of other trigonometric ratios.  Answer Verified
Hint:Consider a right angled $\Delta ABC$, right angle at B. Use Pythagoras theorem then find the values of $\cos \theta$, $\tan \theta$ and their reciprocal $\csc \theta$, $\sec \theta$ and $\cot \theta$.

Complete step-by-step answer:
Let us consider a right angled triangle ABC. We know the Pythagoras theorem, also known as Pythagoras theorem; it is a fundamental relation in Euclidean geometry among the 3 sides of a right triangle. It states that the area of the squares whose sides is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
$A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}$
We have been given, $\sin \theta =\dfrac{3}{5}$.
In the triangle ABC, $\angle B=90{}^\circ$
And take $\angle C=\theta$
Here, $\sin \theta =$opposite side/hypotenuse = $\dfrac{AB}{BC}$
$\cos \theta =$Adjacent side/hypotenuse = $\dfrac{BC}{AC}$
Given,$\sin \theta =\dfrac{3}{5}$
AB=3 and AC=5
Using the Pythagoras theorem, $A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}$
\begin{align} & {{5}^{2}}={{3}^{2}}+B{{C}^{2}} \\ & \Rightarrow B{{C}^{2}}={{5}^{2}}-{{3}^{2}} \\ & \Rightarrow BC=\sqrt{25-9}=\sqrt{16}=4 \\ & \therefore \cos \theta =\dfrac{4}{5} \\ \end{align}
$\tan \theta =$Opposite side/adjacent side=$\dfrac{AB}{BC}=\dfrac{3}{4}$
$\cos ec\theta =\dfrac{1}{\sin \theta }=\dfrac{1}{\dfrac{3}{5}}=\dfrac{5}{3}$
\begin{align} & \sec \theta =\dfrac{1}{\cos \theta }=\dfrac{1}{\dfrac{4}{5}}=\dfrac{5}{4} \\ & \cot \theta =\dfrac{1}{\tan \theta }=\dfrac{1}{\dfrac{3}{4}}=\dfrac{4}{3} \\ & \therefore \sin \theta =\dfrac{3}{5},\cos \theta =\dfrac{4}{5},\tan \theta =\dfrac{3}{4} \\ & \cos ec\theta =\dfrac{5}{3},\sec \theta =\dfrac{5}{4},\cot \theta =\dfrac{4}{3} \\ \end{align}

Note: There are three types of special right triangle, 30-60-90 triangle, 45-45-90 triangle and Pythagoras triple triangles.

Bookmark added to your notes.
View Notes
Sin Cos Tan Values  Sin Theta Formula  Log Values From 1 to 10  Trigonometry Values  Some Application of Trigonometry for Class 10  Trigonometry for Class 10  Absolute Values  Eigen Values  Other Forms of Audit  CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry Formula  