If sin $A = 3/5$ and A is an acute angle, find the values of cos A and tan A.

Step 1: Use the trigonometric identity

We are given the value of $\sin A$ and we need to find $\cos A$. The fundamental trigonometric identity $\sin^2 A + \cos^2 A = 1$ relates these two quantities, so we will use it to find the value of $\cos A$.

Step 2: Substitute the value of sin A

Substitute the given value $\sin A = \frac{3}{5}$ into the identity. Squaring $\frac{3}{5}$ gives $\frac{9}{25}$.

Step 3: Simplify and solve for cos A

Subtract $\frac{9}{25}$ from both sides to isolate $\cos^2 A$. Then, find a common denominator and subtract the fractions. Finally, take the square root of both sides. Since $A$ is an acute angle (between $0^\circ$ and $90^\circ$), $\cos A$ must be positive.

Step 4: Calculate tan A

Now that we have both $\sin A$ and $\cos A$, we can find $\tan A$ using its definition as the ratio of $\sin A$ to $\cos A$. This is another fundamental trigonometric identity.

Step 5: Substitute values of sin A and cos A

Substitute the given value of $\sin A = \frac{3}{5}$ and the calculated value of $\cos A = \frac{4}{5}$ into the formula for $\tan A$. Simplify the complex fraction by multiplying by the reciprocal of the denominator.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
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• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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