Exam-style CBSE practice questions on Trigonometric Ratios of Some Specific Angles (INTRODUCTION TO TRIGONOMETRY). Try each one first, then reveal the correct answer and a step-by-step explanation. Free, from EduLevel — the AI teacher for CBSE.
Given that α + β = 90° and α = 2β, find the value of cos²α + sin²β.
Recall the trigonometric identity relating the squares of sine and cosine of an angle.
Explanation: Substituting α = 2β into the first equation gives 3β = 90°, so β = 30° and α = 60°. The expression becomes cos²(60°) + sin²(30°) = (1/2)² + (1/2)² = 1/4 + 1/4 = 1/2.
If x multiplied by the expression (2tan30°)/(1+tan²30°) is equal to y multiplied by the expression (2tan30°)/(1-tan²30°), determine the ratio x : y.
Recognize the trigonometric identities that resemble the expressions involving tan 30°.
Explanation: The expressions are equivalent to sin(60°) and tan(60°). The equation becomes x(√3/2) = y(√3), which simplifies to x/y = 2/1. Thus, the ratio x : y is 2 : 1.
If tan(3θ) = √3, what is the value of θ/2?
Recall the standard values of trigonometric ratios for common angles; specifically, what angle has a tangent value of √3?
Explanation: Since tan(60°) = √3, we have 3θ = 60°, which gives θ = 20°. Therefore, θ/2 = 20°/2 = 10°.
If sin(4θ) = √3/2, what is the value of θ/3?
Recall the standard values of the sine function for common angles. What angle has a sine value of √3/2?
Explanation: Since sin(60°) = √3/2, we have 4θ = 60°, which gives θ = 15°. Therefore, θ/3 = 15°/3 = 5°.
If sin α = √3/2 and cos β = √3/2, what is the value of the product tan α tan β?
Recall the values of trigonometric ratios for standard angles like 30°, 45°, and 60° to identify the angles α and β.
Explanation: Given sin α = √3/2, the angle α is 60°. Given cos β = √3/2, the angle β is 30°. Therefore, tan α tan β = tan 60° tan 30° = (√3) * (1/√3) = 1.
Calculate the value of the expression 5/8 sec²60° - tan²60° + cos²45°.
This question involves evaluating a trigonometric expression. Recall the standard values of trigonometric functions for common angles like 60° and 45°.
Explanation: Substituting the standard trigonometric values sec 60° = 2, tan 60° = √3, and cos 45° = 1/√2 into the expression and simplifying leads to the result 0.
If cos θ = √3/2, find the value of the expression (cosec²θ - sec²θ) / (cosec²θ + sec²θ).
Recall the relationship between cosine and other trigonometric ratios like secant, cosecant, and tangent. Also, remember the values of trigonometric functions for standard angles.
Explanation: Given cos θ = √3/2, the angle θ is 30°. Calculating cosec 30° = 2 and sec 30° = 2/√3, and substituting these values into the expression yields 1/2.
Determine the value of θ for which the equation 2 sin(2θ) = 1 holds true.
Start by isolating the trigonometric function, sin(2θ), on one side of the equation.
Explanation: Given 2 sin(2θ) = 1, we get sin(2θ) = 1/2. Since sin(30°) = 1/2, we have 2θ = 30°, which implies θ = 15°.
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