Question
Question: If \[\cos A = \dfrac{9}{{41}}\], find \[\tan A\] and \[\csc A\]....
If cosA=419, find
tanA and cscA.
Solution
Here, we need to find the value of tanA and cscA. We will use the trigonometric identities and ratios to solve the question. We will use the formula sin2θ+cos2θ=1 to find the value of sine of A. Using the value of sine of A, we will find the value of cosecant of A. Using the value of sine and cosine of A, we can find the value of the tangent of A.
Formula Used: The sum of squares of the sine and cosine of an angle is equal to 1, that is sin2θ+cos2θ=1.
The tangent of an angle θ is the ratio of the sine and cosine of the angle θ, that is tanθ=cosθsinθ.
The cosecant of an angle θ is the reciprocal of the sine of the angle θ, that is cscθ=sinθ1.
Complete step-by-step answer:
We can find the value of tanA and cscA if we have the values of sinA and cosA.
First, we will find the value of sine of angle of A.
As we know sin2θ+cos2θ=1.
Therefore, we can write sin2A+cos2A=1.
Substituting the given value cosA=419, we get
⇒sin2A+(419)2=1
Simplifying the expression, we get
⇒sin2A+168181=1
Subtracting 168181 from both sides of the equation, we get
⇒sin2A+168181−168181=1−168181 ⇒sin2A=16811681−81 ⇒sin2A=16811600
Taking the square root on both the sides, we get
⇒sinA=16811600 ⇒sinA=4140
Now, we can find the value of cscA.
The cosecant of an angle θ is the reciprocal of the sine of the angle θ, that is cscθ=sinθ1.
Therefore, we get
cscA=sinA1
Substituting the value sinA=4140, we get
⇒cscA=41401
Simplifying the expression, we get
⇒cscA=4041
Next, we can find the value of tanA.
The tangent of an angle
θ is the ratio of the sine and cosine of the angle θ, that is tanθ=cosθsinθ.
Therefore, we get
tanA=cosAsinA
Substituting the value sinA=4140 and cosA=419, we get
⇒tanA=4194140
Simplifying the expression, we get
∴tanA=940
Therefore, the value of tanA and
cscA is 940 and 4041 respectively.
Note: We can also find the values of tanA and cscA using the definitions of the trigonometric ratios.
We know that cosine of an angle θ in a right angled triangle is given by cosθ=HypotenuseBase.
Since cosA=419, assume that base =9x and hypotenuse =41x.
Using the Pythagoras’s theorem, we get
(Hypotenuse)2=(Base)2+(Perpendicular)2 ⇒(41x)2=(9x)2+(Perpendicular)2
Solving the above equation to find the perpendicular, we get
⇒1681x2=81x2+(Perpendicular)2 ⇒(Perpendicular)2=1681x2−81x2 ⇒(Perpendicular)2=1600x2
Taking square root of both sides, we get
⇒Perpendicular=40x
Now, we know that the cosecant of an angle θ in a right angled triangle is given by cscθ=PerpendicularHypotenuse.
Substituting the values of the hypotenuse and perpendicular, we get
⇒cscA=40x41x=4041
Also, we know that the tangent of an angle θ in a right angled triangle is given by tanθ=BasePerpendicular.
Substituting the values of the hypotenuse and perpendicular, we get
⇒tanA=9x40x=940
Therefore, the value of tanA and cscA is 940 and 4041 respectively.