If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).

Verified step by step guidance

1. Recall the co-function identity for sine and cosine: \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta\).
2. Since \(\csc x = \frac{1}{\sin x}\), express \(\csc\left(\frac{\pi}{2} - \theta\right)\) as \(\frac{1}{\sin\left(\frac{\pi}{2} - \theta\right)}\).
3. Substitute the co-function identity into the expression: \(\csc\left(\frac{\pi}{2} - \theta\right) = \frac{1}{\cos\theta}\).
4. Use the given value \(\cos\theta = \frac{1}{3}\) to rewrite the expression as \(\csc\left(\frac{\pi}{2} - \theta\right) = \frac{1}{\frac{1}{3}}\).
5. Simplify the fraction to find the expression for \(\csc\left(\frac{\pi}{2} - \theta\right)\) in terms of the given cosine value.

Key Concepts

Complementary Angles in Trigonometry

Complementary angles are two angles whose measures add up to 90° (or π/2 radians). In trigonometry, the sine of an angle equals the cosine of its complement, i.e., sin(π/2 - θ) = cos θ. This relationship helps simplify expressions involving complementary angles.

Reciprocal Trigonometric Functions

Reciprocal functions are the inverses of the basic trigonometric functions. For example, cosecant (csc) is the reciprocal of sine, defined as csc θ = 1/sin θ. Understanding this helps in converting between sine and cosecant values to solve problems.

Using Given Values to Find Trigonometric Ratios

Given a trigonometric value like cos θ = 1/3, you can find related ratios using identities. Since sin(π/2 - θ) = cos θ, knowing cos θ allows direct calculation of sin(π/2 - θ), and thus csc(π/2 - θ) by taking the reciprocal. This approach simplifies problem-solving.