If $5 \sin \theta = 4$, then the value of $\frac{\sec \theta +... | Mathematics MCQ | Examoogle

Examoogle • User • info@examoogle.com • EE-2024-8821

Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked

Back to Practice Questions

MathematicsTrigonometry

Prev Next

If $5 \sin \theta = 4$ , then the value of $\frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta}$ is:

Correct Answer

📐 MA • Math Trigonometric Identity SubstitutionMathematicsTrigonometry

Option B

Quick Summary:

Given $5 \sin \theta = 4$ , we have $\sin \theta = \frac{4}{5}$ . In a 3-4-5 triangle, $\cos \theta = \frac{3}{5}$ . The expression simplifies to $(\sec \theta + \tan \theta)^2 = (\frac{1+\sin \theta}{\cos \theta})^2 = (\frac{1+4/5}{3/5})^2 = (\frac{9/5}{3/5})^2 = 3^2 = 9$ .

📐MAMath SolutionTrigonometric Identity Substitution

📋 Given

5 sin(theta) = 4, where theta is an angle in a right-angled triangle.

🔢 Formula Used

sin⁡2θ+cos⁡2θ=1,sec⁡θ=1cos⁡θ,tan⁡θ=sin⁡θcos⁡θ\sin^2 \theta + \cos^2 \theta = 1, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta}sin2θ+cos2θ=1,secθ=cosθ1​,tanθ=cosθsinθ​

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often try to calculate the angle theta instead of using the ratio property of the triangle, leading to unnecessary complexity.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Find the sine and cosine values

From $5 \sin \theta = 4$ , we get $\sin \theta = \frac{4}{5}$ . Using the Pythagorean identity $\cos \theta = \sqrt{1 - \sin^2 \theta}$ , we calculate $\cos \theta = \sqrt{1 - (\frac{4}{5})^2} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5}$ .

sin⁡θ=45,cos⁡θ=35\sin \theta = \frac{4}{5}, \quad \cos \theta = \frac{3}{5}sinθ=54​,cosθ=53​

Express the expression in terms of sin and cos

Convert $\sec \theta$ and $\tan \theta$ into sine and cosine functions: $\frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{\frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta}}$ .

1+sin⁡θ1−sin⁡θ\frac{1 + \sin \theta}{1 - \sin \theta}1−sinθ1+sinθ​

Substitute the known values

Substitute $\sin \theta = \frac{4}{5}$ into the simplified expression $\frac{1 + \sin \theta}{1 - \sin \theta}$ .

1+451−45=9515=9\frac{1 + \frac{4}{5}}{1 - \frac{4}{5}} = \frac{\frac{9}{5}}{\frac{1}{5}} = 91−54​1+54​​=51​59​​=9

✅

B is correct because the substitution of $\sin \theta = \frac{4}{5}$ into the expression yields a final value of 9.

Core Concepts Used

Click any tag to open in AI Tutor

Trigonometric ratios Pythagorean identity Algebraic simplification

💡 EXAM TIP

This concept is a precursor to solving trigonometric equations and height-distance problems where calculating ratios is more efficient than finding angles.

Discussion (0)

Loading discussion...

Prev Next

Correct Answer

Related Questions

Discussion (0)