Multiple Choice Questions
Trigonometry: Applications and Graphs
Topic: Trigonometry: Applications and Graphs
Grade: 10
Question 1:
What is the value of sin(45°) + cos(45°)?
Answer choices:
A) 1
B) √2
C) √3/2
D) 1/√2
Answer: B) √2
Explanation:
The value of sin(45°) is 1/√2 and the value of cos(45°) is 1/√2. Adding them together gives √2/√2 which simplifies to √2.
Example:
Consider a right-angled triangle with one of the angles being 45°. If the length of the hypotenuse is 1, then the length of both the adjacent and opposite sides would be 1/√2. Therefore, sin(45°) + cos(45°) = 1/√2 + 1/√2 = √2/√2 = √2.
Question 2:
What is the value of tan(60°) – cot(30°)?
Answer choices:
A) √3
B) 1/√3
C) 1
D) √3/3
Answer: B) 1/√3
Explanation:
The value of tan(60°) is √3 and the value of cot(30°) is √3/3. Subtracting them gives √3 – √3/3, which can be simplified to (√3 – 1)/√3. Rationalizing the denominator gives (3 – √3)/3√3. Simplifying further gives 1/√3.
Example:
Consider an equilateral triangle with each angle measuring 60°. The length of the side would be 1. The value of tan(60°) can be found by dividing the length of the opposite side by the length of the adjacent side, which gives √3/1 = √3. The value of cot(30°) can be found by dividing the length of the adjacent side by the length of the opposite side, which gives 1/(1/√3) = √3/3. Therefore, tan(60°) – cot(30°) = √3 – √3/3 = (√3 – 1)/√3 = 1/√3.
Question 3:
What is the period of the function y = 2sin(3x)?
Answer choices:
A) π/3
B) 2Ï€/3
C) π
D) 2Ï€/3
Answer: C) π
Explanation:
The period of a function y = a*sin(bx) is given by 2Ï€/b. In this case, the coefficient of x is 3, so the period is 2Ï€/3.
Example:
Consider the function y = 2sin(x). The graph of this function completes one full cycle from 0 to 2Ï€. Now, if we multiply the x values by 3, the graph will complete one full cycle in a smaller interval, from 0 to 2Ï€/3. Therefore, the period of y = 2sin(3x) is 2Ï€/3.
Question 4:
What is the value of cos(Ï€/3)?
Answer choices:
A) 1/2
B) √3/2
C) 1
D) √2/2
Answer: B) √3/2
Explanation:
The value of cos(π/3) can be found by considering a 30-60-90 triangle. In this triangle, the length of the hypotenuse is 1, the length of the side opposite to the 60° angle is √3, and the length of the side adjacent to the 60° angle is 1/2. Therefore, cos(π/3) = 1/2.
Example:
Consider an equilateral triangle with each angle measuring 60°. The length of each side is 1. If we draw an altitude from one vertex to the opposite side, we get a right-angled triangle with a 30° angle. The length of the side opposite to the 30° angle is 1/2, the length of the side adjacent to the 30° angle is √3/2, and the length of the hypotenuse is 1. Therefore, cos(π/3) = √3/2.
Question 5:
What is the value of tan(Ï€/4) – sin(Ï€/4)?
Answer choices:
A) 1
B) √2
C) √2/2
D) 1/√2
Answer: C) √2/2
Explanation:
The value of tan(Ï€/4) is 1 and the value of sin(Ï€/4) is 1/√2. Subtracting them gives 1 – 1/√2, which can be simplified to (√2 – 1)/√2. Rationalizing the denominator gives (√2 – 1)/(√2 * √2) = (√2 – 1)/2. Simplifying further gives √2/2 – 1/2 = √2/2 – 1/2 = (√2 – 1)/2.
Example:
Consider a right-angled triangle with one of the angles being 45°. If the length of the hypotenuse is 1, then the length of both the adjacent and opposite sides would be 1/√2. Therefore, tan(Ï€/4) = 1 and sin(Ï€/4) = 1/√2. Subtracting them gives 1 – 1/√2 = (√2 – 1)/√2. Rationalizing the denominator gives (√2 – 1)/(√2 * √2) = (√2 – 1)/2 = √2/2 – 1/2 = (√2 – 1)/2.
Note: More questions and explanations can be provided upon request.