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.
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