Trigonometry If $\sin \theta = \frac{5}{13}$ and θ is acute, find: * cos θ * tan θ

Step 1: Understand the given information

We are given the value of $\sin \theta$ for an acute angle $\theta$. We need to find the values of $\cos \theta$ and $\tan \theta$. We can use the fundamental trigonometric identity relating sine and cosine.

Step 2: Use the Pythagorean Identity to find cos θ

The fundamental trigonometric identity states that $\sin^2 \theta + \cos^2 \theta = 1$. By substituting the given value of $\sin \theta$ into this identity, we can solve for $\cos \theta$. Since $\theta$ is an acute angle (between $0^\circ$ and $90^\circ$), its cosine value must be positive.

Step 3: Find tan θ using sin θ and cos θ

The tangent of an angle is defined as the ratio of its sine to its cosine. Now that we have both $\sin \theta$ and $\cos \theta$, we can easily calculate $\tan \theta$.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• 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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