For the following exercises, find the exact value of each expression.

For the following exercises, find the exact value of each expression. sec 𝜋/6 cot 𝜋/4

The Correct Answer and Explanation is:

To find the exact value of the expression:

sec(π/6) × cot(π/4)

We first evaluate each trigonometric function individually using known exact values from the unit circle.

Step 1: Evaluate sec(π/6)

We know that:cos⁡(π/6)=32\cos(\pi/6) = \frac{\sqrt{3}}{2}cos(π/6)=23​​

Since secant is the reciprocal of cosine:sec⁡(π/6)=1cos⁡(π/6)=13/2=23\sec(\pi/6) = \frac{1}{\cos(\pi/6)} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}sec(π/6)=cos(π/6)1​=3​/21​=3​2​

Now rationalize the denominator:sec⁡(π/6)=23⋅33=233\sec(\pi/6) = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}sec(π/6)=3​2​⋅3​3​​=323​​

Step 2: Evaluate cot(π/4)

Cotangent is the reciprocal of tangent. From the unit circle:tan⁡(π/4)=1⇒cot⁡(π/4)=1tan⁡(π/4)=1\tan(\pi/4) = 1 \Rightarrow \cot(\pi/4) = \frac{1}{\tan(\pi/4)} = 1tan(π/4)=1⇒cot(π/4)=tan(π/4)1​=1

Step 3: Multiply the results

sec⁡(π/6)⋅cot⁡(π/4)=233⋅1=233\sec(\pi/6) \cdot \cot(\pi/4) = \frac{2\sqrt{3}}{3} \cdot 1 = \frac{2\sqrt{3}}{3}sec(π/6)⋅cot(π/4)=323​​⋅1=323​​

Final Answer:

233\boxed{\frac{2\sqrt{3}}{3}}323​​​

Explanation

Trigonometric functions such as secant and cotangent are based on ratios derived from the unit circle. The angle π/6 corresponds to 30 degrees, and π/4 corresponds to 45 degrees. These angles are commonly used because their sine, cosine, tangent, and related functions yield simple values.

To begin, secant is the reciprocal of cosine. At π/6, the cosine is √3 divided by 2. Taking the reciprocal gives 2 over √3. Since it is standard to express answers with rational denominators, we multiply both numerator and denominator by √3, which results in 2√3 over 3.

Next, cotangent is the reciprocal of tangent. At π/4, the tangent is 1, so cotangent is also 1. This is because sine and cosine are equal at π/4, and tangent is sine over cosine.

The final expression is a product of two exact values: sec(π/6) times cot(π/4). Multiplying a number by 1 does not change its value. So, the result of this product is simply the value we obtained for sec(π/6), which is 2√3 over 3.

This method highlights how important it is to understand the relationships between trigonometric functions and to be comfortable working with fractions and radicals. Understanding the unit circle and knowing exact values at key angles are critical skills in trigonometry. These foundations are also useful when solving more complex problems in calculus, physics, and engineering.