Trigonometry: Calculate Cosecant from Cosine
We are given the value of cosine of an angle A and need to find the value of its cosecant.
Given: $\cos(A) = \frac{7}{25}$
Using Trigonometric Identities
We can use the fundamental Pythagorean identity:
\[ \sin^2(A) + \cos^2(A) = 1 \]Substitute the given value of $\cos(A)$:
\[ \sin^2(A) + \left(\frac{7}{25}\right)^2 = 1 \]Calculate the square:
\[ \sin^2(A) + \frac{49}{625} = 1 \]Isolate $\sin^2(A)$:
\[ \sin^2(A) = 1 - \frac{49}{625} \] \[ \sin^2(A) = \frac{625 - 49}{625} \] \[ \sin^2(A) = \frac{576}{625} \]Find $\sin(A)$ by taking the square root:
Note: Assuming angle A is in the first quadrant, where sine is positive.
\[ \sin(A) = \sqrt{\frac{576}{625}} = \frac{24}{25} \]Calculate Cosecant
The definition of cosecant is the reciprocal of sine:
\[ \text{cosec}(A) = \frac{1}{\sin(A)} \]Substitute the calculated value of $\sin(A)$:
\[ \text{cosec}(A) = \frac{1}{\frac{24}{25}} \] \[ \text{cosec}(A) = \frac{25}{24} \]Alternative Method: Right-Angled Triangle
Consider a right-angled triangle where angle A is one of the acute angles.
We know $\cos(A) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$. So, let Adjacent = 7 and Hypotenuse = 25.
Using the Pythagorean theorem (Opposite$^2$ + Adjacent$^2$ = Hypotenuse$^2$):
Opposite$^2 + 7^2 = 25^2$
Opposite$^2 + 49 = 625$
Opposite$^2 = 625 - 49 = 576$
Opposite = $\sqrt{576} = 24$
Now, calculate cosecant: $\text{cosec}(A) = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{25}{24}$.
Both methods confirm the result.
Similar Questions
- If $\sin \theta = \cos \theta$, what is the value of $\theta$ (in degrees)?
- Find the value of $\sin^2 90^\circ + \cos^2 60^\circ$.
- If $cotA = x + \frac{1}{x}$, then what is $cosec^2A$?
- If $secA + tanA = \frac{5}{2}$, then what is $secA - tanA$?
- If $\sin A + \cos A = 1$, then find the value of $\sin^4 A + \cos^4 A$.
- If $\sin A = \frac{4}{5}$ and A is acute, find $\cot(90^\circ - A)$.
If $\sin A = \frac{2}{3}$, find the value of $(3\sin A - 4\cos A)^2 + (4\sin A + 3\cos A)^2$.
- If $\sin A + \cos A = \sqrt{2}$, then find $\sin^3 A + \cos^3 A$.
- If $\sin x = \cos(5x - 60^\circ)$, then what is the value of x?
- If $sinA = \frac{m}{n}$ and A ∈ (0, π/2), then what is $cosA$ in terms of m and n?
Important Questions from Trigonometric Ratios and Identities
If cosec θ = 13/12, then sin θ + cos θ - tan θ is equal to:
What is the value of \(\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}\) ?
If \(\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{k + 1}}{{k - 1}},\) then k = ?
If α + β = 90° and α = 2β, then the value of 3 cos 2 α - 2 sin 2 β is equal to:
If \(\sqrt{3}\) tan θ = 3 sin θ, then what is the value of sin 2θ − cos 2θ ?
