What is the exact value of sin 30° ? a) 1/2 b) \sqrt{3}/2 c) \sqrt{2}/2 d) \sqrt{3} 2. Find the value of side y to the nearest unit. a.) 20 b.) 23 c.) 21 d.) 24 3. Given the figure on the right, which of the following statement 70° is correct? a.) x = 4 b.) sin 70° = \frac{y}{4} c.) tan 70° = \frac{y}{x} d.) cos 70° = \frac{4}{x} 4. In the triangle COV, what is the length of CO? a.) 5.78 in b.) 4.54 in c.) 6.34 in d.) 7.28 in
The Correct Answer and Explanation is:
Here are the correct answers and a detailed explanation for each question.
Correct Answers:
- a) 1/2
- c) 21
- d) cos 70° = 4/x
- c) 6.34 in
Explanation
Here is a step-by-step breakdown of how to arrive at the solution for each problem:
1. What is the exact value of sin 30°?
The value of sine for standard angles like 30°, 45°, and 60° is a fundamental concept in trigonometry. For a 30° angle in a right-angled triangle (specifically a 30-60-90 triangle), the sides are in a specific ratio of 1:√3:2. The side opposite the 30° angle is the shortest side (1), and the hypotenuse is the longest side (2). The definition of sine (SOH) is the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, sin 30° = Opposite / Hypotenuse = 1/2. This makes option (a) the correct choice.
2. Find the value of side y to the nearest unit.
This problem requires using a trigonometric ratio to find an unknown side length. In the given right triangle, we know an angle (50°) and the length of the side adjacent to it (18). We need to find the length of the side opposite the angle, which is y. The trigonometric function that connects the opposite and adjacent sides is the tangent (TOA: Tangent = Opposite / Adjacent). We set up the equation: tan(50°) = y / 18. To solve for y, we multiply both sides by 18, which gives us y = 18 * tan(50°). Using a calculator, tan(50°) is approximately 1.1918. So, y ≈ 18 * 1.1918 ≈ 21.45. The question asks for the value to the nearest unit, so we round 21.45 to 21, making option (c) correct.
3. Given the figure on the right, which of the following statement is correct?
This question tests the understanding of trigonometric definitions (SOH CAH TOA) in a right triangle. For the given 70° angle, the side opposite is y, the side adjacent is 4, and the hypotenuse is x. Let’s check each statement.
- a) x = 4: This is incorrect. The hypotenuse (x) is always the longest side.
- b) sin 70° = y/4: This is incorrect. Sine is Opposite/Hypotenuse, so sin 70° = y/x.
- c) tan 70° = y/x: This is incorrect. Tangent is Opposite/Adjacent, so tan 70° = y/4.
- d) cos 70° = 4/x: This is correct. Cosine is Adjacent/Hypotenuse, which matches the ratio given. Thus, option (d) is the correct statement.
4. In the triangle COV, what is the length of CO?
In the right triangle COV, we are given angle C = 65° and the hypotenuse CV = 15 in. We need to find the length of side CO, which is the side adjacent to the 65° angle. The cosine function relates the adjacent side and the hypotenuse (CAH: Cosine = Adjacent / Hypotenuse). The equation is cos(65°) = CO / 15. To find the length of CO, we rearrange the formula to CO = 15 * cos(65°). Using a calculator, cos(65°) ≈ 0.4226. Therefore, CO ≈ 15 * 0.4226 ≈ 6.339. Rounding to two decimal places, we get 6.34 in. This matches option (c).thumb_upthumb_down
