MATHEMATICS | MOST PREDICTED QUESTIONS

CHAPTER 5 Trigonometry

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A person standing 80m away from a building observes the top of the building at an angle of elevation of 45°. What is the height of the building?

The height can be found using the tangent function: tan(45°) = height / distance. So, height = 80 * tan(45°) = 80m.

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How is the sine of an angle in the unit circle defined?

In the unit circle, the sine of an angle is defined as the y-coordinate of the point where the terminal side of the angle intersects the circle. This definition helps in understanding the properties of the sine function and its periodic nature, as well as its relationship with other trigonometric functions.

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What is the relationship between the sine and cosine of complementary angles?

The sine of an angle is equal to the cosine of its complementary angle. Thus, sin θ = cos (90° – θ).

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The angle of elevation of the sun decreases from 60° to 30°. How does the length of the shadow of a 10m tall tree change?

When the angle of elevation is 60°, the shadow length is 10 * tan(60°) = 10√3. When the angle of elevation is 30°, the shadow length is 10 * tan(30°) = 10/√3.

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A person stands 200m away from a building and observes the top of the building at an angle of elevation of 30°. What is the height of the building?

The height can be found using the tangent function: tan(30°) = height / distance. So, height = 200 * tan(30°) = 200 * (1/√3) = 200/√3m.

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A street light casts a shadow of 15m when the angle of elevation of the sun is 30°. What is the height of the street light?

The height can be found using the tangent function: tan(30°) = height / shadow length. So, height = 15 * tan(30°) = 15 * (1/√3) = 15/√3m.

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A car is parked 100m away from a building. The top of the building is observed at an angle of elevation of 60°. What is the height of the building?

The height can be found using the tangent function: tan(60°) = height / distance. So, height = 100 * tan(60°) = 100 * √3 = 100√3m.

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A person is standing 150m away from a tree and observes the top of the tree at an angle of elevation of 45°. He walks towards the tree and stops at a point where the angle of elevation is 60°. How much distance did he cover?

Initially, tan(45°) = height / 150m, so height = 150m. After walking a certain distance, tan(60°) = height / x, so height = 50√3. The distance covered = 150 – x = 150 – 50√3m.

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A plane is flying at an altitude of 1000m and an observer on the ground sees it at an angle of elevation of 30°. How far is the plane from the observer?

The distance can be found using the sine function: sin(30°) = height / hypotenuse. So, hypotenuse = 1000 / sin(30°) = 1000 / (1/2) = 2000m. The horizontal distance is 2000 * cos(30°) = 2000 * (√3/2) = 1000√3m.

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A boat is sighted from a lighthouse, making an angle of depression of 30°. If the height of the lighthouse is 60m, what is the distance of the boat from the lighthouse base?

The distance can be found using the tangent function: tan(30°) = height / distance. So, distance = 60 / tan(30°) = 60 / (1/√3) = 60√3m.

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A person standing at a distance of 100m from a tower observes the top of the tower at an angle of elevation of 30°. He walks towards the tower and after covering 50m, he observes the top of the tower at an angle of elevation of 60°. What is the height of the tower?

Initially, tan(30°) = height / 100m, so height = 100tan(30°) = 100/√3. After covering 50m, tan(60°) = height / 50m, so height = 50tan(60°) = 50√3. Therefore, height = 50√3m.

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A ladder is leaning against a wall, forming an angle of 60° with the ground. If the base of the ladder is 3m away from the wall, what is the length of the ladder?

The length of the ladder can be found using the cosine function: cos(60°) = adjacent / hypotenuse. So, hypotenuse = 3 / cos(60°) = 3 / (1/2) = 3 * 2 = 6m.

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What is the cosine of 60°?

The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the hypotenuse. For 60°, this value is 0.5.

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An observer is 100m from a building and observes the top of the building with an angle of elevation of 30°. What is the height of the building?

The height can be found using the tangent function: tan(30°) = height / distance. So, height = 100 * tan(30°) = 100 * (1/√3) = 100/√3m.

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A plane flying at an altitude of 2000m is observed at an angle of elevation of 30°. After 1 minute, the angle of elevation changes to 45°. What is the speed of the plane?

Initially, the horizontal distance is 2000/√3m. After 1 minute, the distance becomes 2000m. The difference in distance is 2000 – 2000/√3 = 2000(√3 – 1)/√3. Therefore, speed = 2000(√3 – 1)/√3 = 1000m/min.

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A kite is flying at a height of 100m. If the string makes an angle of 30° with the ground, what is the length of the string?

The length of the string can be found using the sine function: sin(30°) = height / hypotenuse. So, string length = 100 / sin(30°) = 100 / (1/2) = 200m.

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Which trigonometric function is undefined for an angle of 90°?

The tangent function is undefined for an angle of 90° because it involves dividing by the cosine of the angle, which is zero at 90°. This leads to a division by zero, making the tangent function approach infinity and thus undefined at this angle.

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What is the value of tan(45°)?

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the adjacent side. For 45°, this value is 1.

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A tree casts a shadow 30m long. If the angle of elevation of the sun is 45°, what is the height of the tree?

The height can be found using the tangent function: tan(45°) = height / shadow length. So, height = 30 * tan(45°) = 30m.

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A street light casts a shadow 12m long when the angle of elevation of the sun is 30°. What is the height of the street light?

The height can be found using the tangent function: tan(30°) = height / shadow length. So, height = 12 * tan(30°) = 12 / √3 = 12/√3m.

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What is the measure of the angle of depression if the angle of elevation is 35°?

The angle of depression is equal to the angle of elevation. Thus, if the angle of elevation is 35°, the angle of depression is also 35°.

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A building casts a shadow of 50m when the angle of elevation of the sun is 60°. What is the height of the building?

The height can be found using the tangent function: tan(60°) = height / shadow length. So, height = 50 * tan(60°) = 50 * √3 = 50√3m.

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A tree is 30m tall. The angle of elevation from a point 50m away from the base of the tree is?

The angle of elevation can be found using the tangent function: tan(θ) = height / distance. So, tan(θ) = 30 / 50 = 3/5. Using a calculator, θ ≈ 30°.

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What is the value of cos(0°)?

The value of cos(0°) is 1. This is because, in a right-angled triangle, the adjacent side is equal to the hypotenuse when the angle is 0°, making the ratio (adjacent/hypotenuse) equal to 1. This value is a fundamental property of the cosine function and is used in various trigonometric calculations.

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A ship is sighted at an angle of depression of 20° from the top of a 200m lighthouse. How far is the ship from the lighthouse?

The distance can be found using the tangent function: tan(20°) = height / distance. So, distance = 200 / tan(20°) ≈ 584m.

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What is the length of the hypotenuse in a right-angled triangle with sides 6 and 8?

Using the Pythagorean theorem, the hypotenuse is √(6² + 8²) = √(36 + 64) = √100 = 10.

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A ladder leans against a wall forming an angle of 60° with the ground. If the length of the ladder is 10m, what is the height reached on the wall?

The height can be found using the sine function: sin(60°) = height / hypotenuse. So, height = 10 * sin(60°) = 10 * (√3/2) = 5√3 m.

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A kite is flying at a height of 80m above the ground. The string attached to the kite makes an angle of 60° with the ground. What is the length of the string?

The length of the string can be found using the sine function: sin(60°) = height / hypotenuse. So, hypotenuse = 80 / sin(60°) = 80 / (√3/2) = 80 * 2/√3 = 80√3m.

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What is the value of sin(30°)?

The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the hypotenuse. For 30°, this value is 0.5.

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What is the degree measure of an angle that is π/6 radians?

To convert radians to degrees, multiply by 180/π. Thus, π/6 radians equals 30°.

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A ramp is inclined at an angle of 30° to the horizontal. If the length of the ramp is 10m, what is the height of the ramp?

The height can be found using the sine function: sin(30°) = height / hypotenuse. So, height = 10 * sin(30°) = 10 * (1/2) = 5m.

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How is the angle in radians defined in the unit circle?

In the unit circle, an angle in radians is defined as the arc length divided by the radius. For a unit circle (radius = 1), this simplifies to the arc length itself. This definition is crucial for converting between degrees and radians and understanding the relationship between angles and circular motion.

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A person is standing at the edge of a cliff 100m high. The angle of depression to a boat on the water is 30°. How far is the boat from the base of the cliff?

The distance can be found using the tangent function: tan(30°) = height / distance. So, distance = 100 / tan(30°) = 100 / (1/√3) = 100√3m.

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How is the angle of elevation defined in trigonometry?

The angle of elevation is defined as the angle between the horizontal and the line of sight when an observer looks at an object above the horizontal level. This angle is crucial in practical applications such as determining heights and distances in various fields, including navigation and surveying.

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What is the Pythagorean identity in trigonometry?

The Pythagorean identity in trigonometry states that sin²(θ) + cos²(θ) = 1 for any angle θ. This identity is derived from the Pythagorean theorem and is fundamental in proving other trigonometric identities and solving equations involving trigonometric functions.

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A helicopter is flying at a height of 150m. If the angle of elevation from a point 50m away horizontally is observed, what is the angle of elevation?

The angle of elevation can be found using the tangent function: tan(θ) = height / distance. So, tan(θ) = 150 / 50 = 3. Using a calculator, θ ≈ 75°.

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What is the tangent of an angle in a right-angled triangle?

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the adjacent side. It is an essential function in trigonometry, used to relate the angles and sides of a triangle, particularly in solving problems involving right-angled triangles.

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A drone is flying at a height of 200m and is observed at an angle of elevation of 45° from a point on the ground. After 10 seconds, the angle of elevation changes to 30°. Assuming the drone is moving horizontally, what is the horizontal distance covered by the drone in 10 seconds?

Initially, the drone is 200m above the ground, and the distance from the observer is 200m (since tan(45°) = 1). When the angle of elevation changes to 30°, the height remains the same. The new horizontal distance is 200/√3. Therefore, the horizontal distance covered = (200/√3) – 200 = 200(√3-1)m.

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What is the period of the sine function?

The sine function has a period of 360°, meaning it repeats its values every 360°. This periodicity is a fundamental characteristic of trigonometric functions, allowing them to model periodic phenomena such as waves and oscillations effectively.

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What is the relationship between the secant function and the cosine function?

The secant function is the reciprocal of the cosine function. It is defined as sec(θ) = 1/cos(θ) for any angle θ where the cosine is not zero. This relationship is used in various trigonometric identities and equations, providing a different perspective on the behavior and properties of trigonometric functions.

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Which trigonometric function is used to find the height of a building when the angle of elevation and distance from the building are known?

The tangent function is used to find the height of a building when the angle of elevation and the horizontal distance from the building are known. The tangent of the angle of elevation equals the ratio of the building’s height to the horizontal distance, allowing for the calculation of height using the formula tan(θ) = height/distance.

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A person stands 50m away from a building. The angle of elevation to the top of the building is 60°. What is the height of the building?

The height can be found using the tangent function: tan(60°) = height / distance. So, height = 50 * tan(60°) = 50 * √3 = 50√3m.

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What is the primary purpose of the sine function in trigonometry?

The sine function in trigonometry is primarily used to relate the angle of a right-angled triangle to the ratio of the opposite side to the hypotenuse. It is a fundamental function for determining the relationship between the angles and lengths of a triangle’s sides.

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An observer on top of a hill 150m high observes a car moving towards the hill. The initial angle of depression of the car is 30°, and after 5 seconds, the angle of depression becomes 60°. What is the speed of the car?

Initially, the horizontal distance is 150/√3. After 5 seconds, the distance becomes 150√3. The difference in horizontal distance is 150√3 – 150/√3 = 150(√3 – 1/√3) = 150(3-1)/√3 = 300/√3. Therefore, speed = 300/√3 / 5 = 30m/s.

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A lighthouse is 100m tall and observes a boat at an angle of depression of 45°. After 5 minutes, the angle of depression changes to 60°. What is the speed of the boat?

Initially, the horizontal distance is 100m (since tan(45°) = 1). After 5 minutes, the distance is 100/√3. The difference in horizontal distance = 100(√3 – 1). Therefore, speed = 100(√3 – 1) / 5 ≈ 15m/min.

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A person looks at the top of a 50m tall tower from a distance of 50m. What is the angle of elevation?

The angle of elevation can be found using the tangent function: tan(θ) = height / distance. So, tan(θ) = 50 / 50 = 1. Therefore, θ = 45°.

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How is the angle of elevation used in real-life applications?

The angle of elevation is primarily used to calculate heights in real-life applications. By measuring the angle of elevation from a known distance, one can determine the height of objects such as buildings, mountains, and other structures using trigonometric functions like tangent.

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A person is looking at the top of a tree from a distance of 50m, making an angle of elevation of 45°. What is the height of the tree?

The height of the tree can be found using the tangent function: tan(45°) = height / distance. So, height = 50 * tan(45°) = 50m.

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A person stands at the top of a 100m high cliff and observes a boat at an angle of depression of 45°. How far is the boat from the base of the cliff?

The distance can be found using the tangent function: tan(45°) = height / distance. So, distance = 100 / tan(45°) = 100m.

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A pole is leaning at an angle of 60° with the ground and its shadow is 5m long. What is the height of the pole?

The height can be found using the tangent function: tan(60°) = height / shadow length. So, height = 5 * tan(60°) = 5√3m.

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How is the cosine of an angle in a right-angled triangle defined?

The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. This function is crucial in solving problems related to right-angled triangles and finding missing sides or angles.

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A person at a height of 50m above the ground observes another person standing 20m away horizontally. What is the angle of depression?

The angle of depression is equal to the angle of elevation. The angle of elevation can be found using the tangent function: tan(θ) = height / distance. So, tan(θ) = 50 / 20 = 5/2. Using a calculator, θ ≈ 45°.

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A flagpole is 20m tall. The shadow of the flagpole is 20m long. What is the angle of elevation of the sun?

The angle of elevation can be found using the tangent function: tan(θ) = height / shadow length. So, tan(θ) = 20 / 20 = 1. Therefore, θ = 45°.

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If a triangle has sides of lengths 3, 4, and 5, what is the measure of the angle opposite the side of length 5?

In a right-angled triangle, the side opposite the right angle is the hypotenuse. For sides 3, 4, and 5, the angle opposite the hypotenuse is 90°.

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What is the amplitude of the cosine function y = 3cos(x)?

The amplitude of a trigonometric function like cosine is the maximum value it reaches. In the function y = 3cos(x), the amplitude is 3, indicating the peak value of the function from its midline, which is a critical parameter in wave analysis.

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A person is observing the top of a hill at an angle of elevation of 60° from a distance of 200m. After climbing 100m vertically, he observes the top of the hill at an angle of elevation of 45°. What is the height of the hill?

Initially, tan(60°) = height / 200m, so height = 200√3. After climbing 100m, tan(45°) = height – 100 / 200, so height – 100 = 200. Therefore, height = 300m.

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Which angle is a right angle in a triangle?

A right angle is an angle of 90°. In a right-angled triangle, one of the angles is always 90°.

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A man standing on the top of a 20m building observes a car parked at a distance, making an angle of depression of 45°. How far is the car from the building?

The distance can be found using the tangent function: tan(45°) = height / distance. So, distance = 20 / tan(45°) = 20m.

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How is the cotangent of an angle in a right-angled triangle defined?

The cotangent of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the opposite side. This function is used in trigonometry to provide an alternative perspective on angle and side relationships, complementing the tangent function.

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A flagpole is 12m tall. The shadow of the flagpole is 12√3m long. What is the angle of elevation of the sun?

The angle of elevation can be found using the tangent function: tan(θ) = height / shadow length. So, tan(θ) = 12 / 12√3 = 1/√3. Using a calculator, θ ≈ 60°.

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What is the angle of depression in trigonometry?

The angle of depression is the angle formed between the horizontal plane and the line of sight when an observer looks downward at an object. This concept is commonly used in navigation, aviation, and engineering to determine the position and distance of objects below the observer’s horizontal line of sight.

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In trigonometry, the sine of an angle is equal to the cosine of its complementary angle. This relationship is expressed as sin(θ) = cos(90° – θ), reflecting the inherent symmetry in trigonometric functions and their applications in various mathematical problems.

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