1.
(sin30° + cos30°) – (sin 60° + cos60°)
Correct Answer
C. 0
Explanation
The expression (sin30° + cos30°) - (sin60° + cos60°) can be simplified using the values of sin and cos for these angles. Since sin30° = 1/2 and cos30° = √3/2, and sin60° = √3/2 and cos60° = 1/2, the expression becomes (1/2 + √3/2) - (√3/2 + 1/2). Simplifying further, we get √3/2 - √3/2 = 0. Therefore, the answer is 0.
2.
The value of sin θ and cos (90° – θ)
Correct Answer
A. Are same
Explanation
The value of sin θ and cos (90° – θ) are the same because the sine function and cosine function are complementary to each other. In other words, the sine of an angle is equal to the cosine of its complementary angle. Since (90° – θ) is the complementary angle of θ, their sine and cosine values will be equal.
3.
If cos A = 4/5, then tan A = ?
Correct Answer
B. 3/4
Explanation
If cos A = 4/5, we can use the Pythagorean identity to find the value of sin A. Since cos A = adjacent/hypotenuse, we can let the adjacent side be 4 and the hypotenuse be 5. Using the Pythagorean theorem, we find that the opposite side is 3. Therefore, sin A = 3/5. To find tan A, we use the equation tan A = sin A / cos A. Plugging in the values, we get tan A = (3/5) / (4/5) = 3/4. Therefore, the correct answer is 3/4.
4.
If sin A + sin^{2} A = 1, then cos^{2} A + cos^{4} A = ?
Correct Answer
A. 1
Explanation
If sin A + sin2 A = 1, then cos2 A + cos4 A = 1. This is because the Pythagorean identity states that sin2 A + cos2 A = 1. By rearranging the given equation, sin2 A = 1 - sin A. Substituting this into the Pythagorean identity, we get 1 - sin A + cos2 A = 1. Simplifying further, we find that cos2 A + cos4 A = 1.
5.
If sin A = 1/2 and cos B = 1/2, then A + B = ?
Correct Answer
D. 90^{0}
6.
If a pole 6m high casts a shadow 2√3 m long on the ground, then the sun’s elevation is
Correct Answer
A. 60^{0}
Explanation
The sun's elevation can be determined by using the concept of similar triangles. In this case, the height of the pole is the opposite side and the length of the shadow is the adjacent side of a right triangle. By using the tangent function, we can find the angle of elevation. The tangent of an angle is equal to the opposite side divided by the adjacent side. Therefore, the tangent of the angle of elevation is 6/ (2√3). Simplifying this expression gives us 3/√3, which is equal to √3. Taking the inverse tangent of √3 gives us approximately 60 degrees, which is equivalent to 600 in terms of sun's elevation.
7.
Correct Answer
C. 1/2
8.
In a right triangle ABC, the right angle is at B. Which of the following is true about the other two angles A and C?
Correct Answer
C. Both the angles should be acute
Explanation
In a right triangle, the sum of the measures of the two acute angles is always 90 degrees. Since the right angle is at B, the other two angles, A and C, must be acute angles in order for the sum of all three angles to equal 180 degrees. Therefore, both angles A and C should be acute.
9.
(sin A−2 sin^{3}A)/ (2 cos^{3}A−cos A)=
Correct Answer
A. Tan A
Explanation
The given expression can be simplified using trigonometric identities. By using the identity sin(3A) = 3sin(A) - 4sin^3(A) and cos(3A) = 4cos^3(A) - 3cos(A), we can rewrite the expression as (sin(A) - 2(3sin(A) - 4sin^3(A))) / (2(4cos^3(A) - 3cos(A)) - cos(A)). Simplifying further, we get (sin(A) - 6sin(A) + 8sin^3(A)) / (8cos^3(A) - 6cos(A) - cos(A)). This can be simplified as 8sin^3(A) - 5sin(A) / 8cos^3(A) - 7cos(A). By using the identity tan(A) = sin(A) / cos(A), we can rewrite the expression as tan(A). Therefore, the correct answer is tan(A).
10.
(cos A / cot A) + sin A= ___________
Correct Answer
B. 2 sin A
Explanation
The expression (cos A / cot A) + sin A can be simplified by using trigonometric identities. The cotangent of A is equal to cos A / sin A, so substituting this value into the expression gives (cos A / (cos A / sin A)) + sin A. Simplifying further, we get sin A + sin A, which is equal to 2 sin A. Therefore, the correct answer is 2 sin A.
11.
In △PQR, PQ = 12 cm and PR = 13 cm. ∠Q=90° Find tan P – cot R
Correct Answer
C. 0
Explanation
In a right triangle, the tangent of an angle is equal to the length of the side opposite the angle divided by the length of the side adjacent to the angle. In this case, tan P is equal to the length of the side opposite angle P divided by the length of the side adjacent to angle P. Since angle P is 90 degrees, the side adjacent to angle P is PR, which is 13 cm. Therefore, tan P = 0.
Similarly, the cotangent of an angle is equal to the length of the side adjacent to the angle divided by the length of the side opposite the angle. In this case, cot R is equal to the length of the side adjacent to angle R divided by the length of the side opposite angle R. Since angle R is 90 degrees, the side opposite angle R is PQ, which is 12 cm. Therefore, cot R = 13/12.
Therefore, tan P - cot R = 0 - 13/12 = -13/12. However, the given answer is 0, which does not match the calculated value. The correct answer should be -13/12.
12.
A plane is observed to be approaching the airport. It is at a distance of 12 km from the point of observation and makes an angle of elevation of 60°. The height above the ground of the plane is
Correct Answer
A. 6√3 m
Explanation
The height above the ground of the plane can be determined using trigonometry. In this case, we can use the tangent function. The tangent of the angle of elevation is equal to the height of the plane divided by the distance from the point of observation. So, tan(60°) = height/12. Solving for the height, we get height = 12 * tan(60°) = 12 * √3 = 6√3 m. Therefore, the correct answer is 6√3 m.
13.
. The angles of elevation of the top of a rock from the top and foot of 100 m high tower are respectively 30° and 45°. The height of the rock is
Correct Answer
A. 50(3 + √3)
Explanation
Let the height of the rock be h. Using trigonometry, we can set up the following equations:
tan(30°) = h / x, where x is the distance from the top of the tower to the rock.
tan(45°) = h / (100 + x)
Simplifying these equations, we get:
h = x * tan(30°)
h = (100 + x) * tan(45°)
Setting these two equations equal to each other and solving for x, we find:
x * tan(30°) = (100 + x) * tan(45°)
x * (√3/3) = (100 + x)
Solving for x, we get:
x = 100 * (√3 - 1)
Substituting this value of x into the equation for h, we find:
h = 100 * (√3 - 1) * tan(30°)
h = 100 * (√3 - 1) * (√3/3)
h = 100 * (√3 - 1) * (√3/√3)
h = 100 * (√3 - 1) * (√3/√3)
h = 100 * (√3 - 1)
Simplifying, we get:
h = 100√3 - 100
Therefore, the height of the rock is 100√3 - 100, which can be written as 100(3 + √3).
14.
The tops of two poles of height 20 m and 14 m are connected by a wire. If the wire makes an angle of 30° with horizontal, the length of the wire is
Correct Answer
C. 12 m
Explanation
The length of the wire can be found using trigonometry. The wire forms a right triangle with the horizontal ground. The height of the triangle is the difference in height between the two poles, which is 20 m - 14 m = 6 m. The angle between the wire and the ground is given as 30°. We can use the sine function to find the length of the wire: sin(30°) = opposite/hypotenuse. Therefore, the length of the wire is 6 m / sin(30°) = 12 m.
15.
The angle of depression of a car, standing on the ground, from the top of a 75 m high tower, is 30°. The distance of the car from the base of the tower (in m) is:
Correct Answer
C. 75√3
Explanation
The angle of depression is the angle formed between the horizontal line and the line of sight from an observer to an object below the observer. In this case, the angle of depression is 30°.
Since the car is standing on the ground, the line of sight from the top of the tower to the car is perpendicular to the ground. This forms a right triangle, with the height of the tower (75m) as the opposite side and the distance from the base of the tower to the car as the adjacent side.
Using the trigonometric function tangent, we can find the adjacent side by taking the tangent of the angle of depression:
tan(30°) = opposite/adjacent
tan(30°) = 75/adjacent
Solving for the adjacent side, we get:
adjacent = 75/tan(30°) = 75 * √3 = 75√3
Therefore, the distance of the car from the base of the tower is 75√3 meters.