Question
Question: If \[\tan ({\cos ^{ - 1}}x) = \sin ({\cot ^{ - 1}}\dfrac{1}{2})\], then find the value of \[x\]. A...
If tan(cos−1x)=sin(cot−121), then find the value of x.
A) x=−35
B) x=+35
C) x=−32
D) x=+32
Solution
Hint: In the given question, we have to apply trigonometric identities to solve the question. We will have to find the hypotenuse of the triangle to arrive at the value of sinθ and then use trigonometric ratios to solve tanθ.
Complete step by step solution:
Let us solve the equation as follows:
First let us consider RHS term sin(cot−121)
Using the trigonometric identity cot−1θ=tan−1θ1, we get,
=sin(tan−12)
Using the formula, tanθ=AdjacentsideOppositeside, we can draw following diagram: (θ is the angle ACB.)
Since tan−1=2, we get
AB=2and
BC=1
Using Pythagoras Theorem (which states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse), we get,
AC=AB2+BC2
AC=(2)2+12
⇒AC=4+1
⇒AC=5
Therefore, we can get value of sinθ,
sinθ=ACAB=52
Hence RHS= 52
Now we can proceed to LHS side of equation tan(cos−1x):
Using the THS result, we will get,
tan(cos−1x)=52
We can write the equation as follows:
cos−1x=tan−152
Let tan−152=t
Therefore, we can say that,
52=tant
Now let us draw the diagram again as follows:
Here tant=B′C′A′B′=52
Using Pythagoras Theorem, we get,
A′C′=A′B′2+B′C′2
A′C′=(2)2+(5)2
A′C′=4+5
A′C′=9
A′C′=3
Therefore, we can get the value of cost=A′C′B′C′
cost=35
Since tan−152=t, we get,
cos−1x=t
Using the property cos−1A=Bso A=cosB, we will get:
x=cost
x=35
Hence, option (B) x=+35 is the correct answer.
Note:
- cot−1θ=tan−1θ1 is proved as below:
Let cot−1(x)=θ
So, we get x=cotθ
We know that cotangent is the inverse of tangent. So, we get,
x1=tanθ
Therefore, we can say that,
θ=tan−1(x1)=cot−1(x)
- Meaning of the terms used to find the missing value in diagram are clarified below:
Sine: The ratio of side opposite to given angle and hypotenuse is called sine. It is denoted as sinθ.
sinθ=HypotenuseSideoppositetogivenangle
In the given sum, we will get sinθ=ACAB
Tangent: The ratio of side opposite to given angle and its adjacent side is called tangent. It is denoted as tanθ.
tanθ=SideadjacenttogivenangleSideoppositetogivenangle
In the given sum, we will get tanθ=B′C′A′B′