Examkin

Question

Difficulty: MediumBasic Trigonometric Ratios, Special Angles, and Identities

If sin⁡θ=45\sin \theta = \frac{4}{5}, where θ\theta is an acute angle, what is the exact value of the expression sec⁡2θ−1cot⁡θ+csc⁡θ\frac{\sec^2 \theta - 1}{\cot \theta + \csc \theta}?

  1. 89\frac{8}{9}Answer
  2. B
    1627\frac{16}{27}
  3. C
    825\frac{8}{25}
  4. D
    43\frac{4}{3}

Answer

The exact value of the expression is 89\frac{8}{9}.
Given an acute angle θ\theta with sin⁡θ=45\sin \theta = \frac{4}{5}, the adjacent side is 33. This gives tan⁡θ=43\tan \theta = \frac{4}{3}, cot⁡θ=34\cot \theta = \frac{3}{4}, and csc⁡θ=54\csc \theta = \frac{5}{4}. Using the identity sec⁡2θ−1=tan⁡2θ\sec^2 \theta - 1 = \tan^2 \theta, the numerator is (43)2=169\left(\frac{4}{3}\right)^2 = \frac{16}{9}. The denominator cot⁡θ+csc⁡θ=34+54=2\cot \theta + \csc \theta = \frac{3}{4} + \frac{5}{4} = 2. Dividing 169\frac{16}{9} by 22 yields 89\frac{8}{9}.

Step-by-Step Solution

1
Determine all required trigonometric ratios for the acute angle θ\theta.
Given sin⁡θ=45\sin \theta = \frac{4}{5}, the opposite side is 44 and the hypotenuse is 55. By the Pythagorean theorem, the adjacent side is 52−42=3\sqrt{5^2 - 4^2} = 3. Therefore, cos⁡θ=35\cos \theta = \frac{3}{5}, tan⁡θ=43\tan \theta = \frac{4}{3}, cot⁡θ=34\cot \theta = \frac{3}{4}, and csc⁡θ=54\csc \theta = \frac{5}{4}.
Defining the right-triangle side lengths allows direct evaluation of all six trigonometric ratios.
2
Simplify the numerator using standard trigonometric identities.
sec⁡2θ−1=tan⁡2θ=(43)2=169\sec^2 \theta - 1 = \tan^2 \theta = \left(\frac{4}{3}\right)^2 = \frac{16}{9}.
Applying the fundamental identity 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta simplifies the numerator.
3
Evaluate the denominator.
cot⁡θ+csc⁡θ=34+54=84=2\cot \theta + \csc \theta = \frac{3}{4} + \frac{5}{4} = \frac{8}{4} = 2.
Adding fractional values with a common denominator.
4
Divide the numerator by the denominator.
16/92=89\frac{16/9}{2} = \frac{8}{9}.
Dividing the simplified numerator by the simplified denominator yields the final result.

Key Concept

Evaluation of trigonometric expressions using fundamental identities and right-triangle ratio definitions.
Rate this question

Topics

Basic Trigonometric Ratios, Special Angles, and Identities

More questions from this topic