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Questions and Answers
A surveyor needs to determine the height of a cliff. From a point 100 meters away from the base of the cliff, the angle of elevation to the top is $60^\circ$. What trigonometric function should they primarily use to find the height?
A surveyor needs to determine the height of a cliff. From a point 100 meters away from the base of the cliff, the angle of elevation to the top is $60^\circ$. What trigonometric function should they primarily use to find the height?
- Tangent (correct)
- Cosecant
- Cosine
- Sine
An airplane is flying at an altitude of 1000 meters. The pilot observes the angle of depression to a landing strip to be $30^\circ$. What is the horizontal distance between the airplane and the landing strip?
An airplane is flying at an altitude of 1000 meters. The pilot observes the angle of depression to a landing strip to be $30^\circ$. What is the horizontal distance between the airplane and the landing strip?
- $2000$ meters
- $1000 / \sqrt{3}$ meters
- $1000$ meters
- $1000 \sqrt{3}$ meters (correct)
A ladder leans against a wall, making an angle of $70^\circ$ with the ground. The foot of the ladder is 2 meters away from the wall. How high up the wall does the ladder reach?
A ladder leans against a wall, making an angle of $70^\circ$ with the ground. The foot of the ladder is 2 meters away from the wall. How high up the wall does the ladder reach?
- $2 \cdot cot(70^\circ)$ meters
- $2 \cdot sin(70^\circ)$ meters
- $2 \cdot cos(70^\circ)$ meters
- $2 \cdot tan(70^\circ)$ meters (correct)
From the top of a cliff 200 meters high, the angle of depression to a boat is $45^\circ$. How far is the boat from the base of the cliff?
From the top of a cliff 200 meters high, the angle of depression to a boat is $45^\circ$. How far is the boat from the base of the cliff?
A kite is flying at the end of a 100-meter string. If the string makes an angle of $60^\circ$ with the ground, how high is the kite above the ground, assuming the string is straight?
A kite is flying at the end of a 100-meter string. If the string makes an angle of $60^\circ$ with the ground, how high is the kite above the ground, assuming the string is straight?
Two ships leave a port at the same time. Ship A sails north at a speed of 20 km/h, and Ship B sails east at a speed of 15 km/h. After 3 hours, how far apart are the two ships?
Two ships leave a port at the same time. Ship A sails north at a speed of 20 km/h, and Ship B sails east at a speed of 15 km/h. After 3 hours, how far apart are the two ships?
A tower stands vertically on the ground. From a point on the ground 20 meters away from the foot of the tower, the angle of elevation to the top of the tower is $45^\circ$. What is the height of the tower?
A tower stands vertically on the ground. From a point on the ground 20 meters away from the foot of the tower, the angle of elevation to the top of the tower is $45^\circ$. What is the height of the tower?
An observer is standing 50 meters away from a building. The angle of elevation to the top of the building is $30^\circ$. What is the height of the building?
An observer is standing 50 meters away from a building. The angle of elevation to the top of the building is $30^\circ$. What is the height of the building?
A person observes the angle of elevation of a mountain peak to be $60^\circ$. The person then walks 500 meters closer and finds the angle of elevation to be $45^\circ$. What equation would be used to compute the height $h$ of the mountain peak?
A person observes the angle of elevation of a mountain peak to be $60^\circ$. The person then walks 500 meters closer and finds the angle of elevation to be $45^\circ$. What equation would be used to compute the height $h$ of the mountain peak?
A ship is sailing towards a lighthouse. The angle of elevation of the top of the lighthouse from the ship is $30^\circ$. After the ship sails 100 meters closer, the angle of elevation becomes $45^\circ$. If $h$ is the height of the lighthouse, which equation is correct?
A ship is sailing towards a lighthouse. The angle of elevation of the top of the lighthouse from the ship is $30^\circ$. After the ship sails 100 meters closer, the angle of elevation becomes $45^\circ$. If $h$ is the height of the lighthouse, which equation is correct?
A 10-meter ladder leans against a building. The angle formed by the ladder and the ground is $60^\circ$. How far is the base of the ladder from the building?
A 10-meter ladder leans against a building. The angle formed by the ladder and the ground is $60^\circ$. How far is the base of the ladder from the building?
From a point on the ground, the angles of elevation to the bottom and top of a flagstaff situated on top of a 20-meter high building are $30^\circ$ and $45^\circ$ respectively. What is the height of the flagstaff?
From a point on the ground, the angles of elevation to the bottom and top of a flagstaff situated on top of a 20-meter high building are $30^\circ$ and $45^\circ$ respectively. What is the height of the flagstaff?
A tree is broken by the wind. The top of the tree struck the ground at an angle of $30^\circ$ and at a distance of 10 meters from the foot. Find the original height of the tree.
A tree is broken by the wind. The top of the tree struck the ground at an angle of $30^\circ$ and at a distance of 10 meters from the foot. Find the original height of the tree.
The angle of elevation of a cloud from a point $h$ meters above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. The height of the cloud above the lake is:
The angle of elevation of a cloud from a point $h$ meters above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. The height of the cloud above the lake is:
From the top of a light house 40 meters high, the angle of depression of a boat is $60^\circ$. What is the distance of the boat from the lighthouse?
From the top of a light house 40 meters high, the angle of depression of a boat is $60^\circ$. What is the distance of the boat from the lighthouse?
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30^\circ$. Which equation relates the height of the tower ($h$) to the distance of the car from the foot of the tower ($d$)?
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30^\circ$. Which equation relates the height of the tower ($h$) to the distance of the car from the foot of the tower ($d$)?
The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is $30^\circ$ than when it is $60^\circ$. What is the height of the tower?
The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is $30^\circ$ than when it is $60^\circ$. What is the height of the tower?
A balloon is connected to a ground observation point by a cable of length 100 m, which makes an angle of $60^\circ$ with the level ground. Assuming that there is no slack in the cable, the height of the balloon is?
A balloon is connected to a ground observation point by a cable of length 100 m, which makes an angle of $60^\circ$ with the level ground. Assuming that there is no slack in the cable, the height of the balloon is?
A man in a boat rowing away from a lighthouse 100 m high takes 2 minutes to change the angle of elevation of the top of the lighthouse from $60^\circ$ to $30^\circ$. The speed of the boat in meters per minute is:
A man in a boat rowing away from a lighthouse 100 m high takes 2 minutes to change the angle of elevation of the top of the lighthouse from $60^\circ$ to $30^\circ$. The speed of the boat in meters per minute is:
Flashcards
Trigonometry
Trigonometry
Branch of math studying triangle side and angle relationships.
Sine (sin θ)
Sine (sin θ)
Ratio of the opposite side to the hypotenuse in a right triangle.
Cosine (cos θ)
Cosine (cos θ)
Ratio of the adjacent side to the hypotenuse in a right triangle.
Tangent (tan θ)
Tangent (tan θ)
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Cosecant (csc θ)
Cosecant (csc θ)
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Secant (sec θ)
Secant (sec θ)
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Cotangent (cot θ)
Cotangent (cot θ)
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Pythagorean Identity
Pythagorean Identity
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Height and Distance Problems
Height and Distance Problems
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Angle of Elevation
Angle of Elevation
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Angle of Depression
Angle of Depression
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Height of Tower
Height of Tower
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Solving Problems
Solving Problems
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Surveying Land
Surveying Land
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Ship-Lighthouse Distance
Ship-Lighthouse Distance
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Navigation
Navigation
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Surveying
Surveying
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Engineering
Engineering
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Astronomy
Astronomy
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Physics
Physics
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Study Notes
- Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles.
- It focuses primarily on right-angled triangles and defines trigonometric functions based on the ratios of the sides.
- Trigonometry is used in various fields like navigation, surveying, engineering, astronomy, and physics.
- The word "trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure).
- The basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).
Basic Trigonometric Functions
- These functions relate an angle of a right triangle to ratios of two of its sides.
- Sine (sin θ) = Opposite / Hypotenuse
- Cosine (cos θ) = Adjacent / Hypotenuse
- Tangent (tan θ) = Opposite / Adjacent = sin θ / cos θ
- Cosecant (csc θ) = 1 / sin θ = Hypotenuse / Opposite
- Secant (sec θ) = 1 / cos θ = Hypotenuse / Adjacent
- Cotangent (cot θ) = 1 / tan θ = Adjacent / Opposite = cos θ / sin θ
Trigonometric Ratios of Standard Angles
- Trigonometric ratios for angles 0°, 30°, 45°, 60°, and 90° are commonly used.
- sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
- cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
- tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = undefined
Trigonometric Identities
- Fundamental equations that are always true for any value of the angle.
- Pythagorean Identities:
- sin² θ + cos² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = csc² θ
- Reciprocal Identities:
- csc θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
- Quotient Identities:
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
Applications of Trigonometry
- Trigonometry finds extensive applications in various fields.
- Height and Distance Problems: Determining heights of towers, buildings, and distances across rivers or inaccessible areas.
- Navigation: Used in maritime and aviation navigation to calculate positions and courses.
- Surveying: Used to determine land boundaries, areas, and elevations.
- Engineering: Essential in structural engineering for calculating forces and stresses in buildings and bridges.
- Astronomy: Used to measure distances to stars and planets.
- Physics: Utilized in wave mechanics, optics, and acoustics.
Angles of Elevation and Depression
- Angle of Elevation: The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.
- Angle of Depression: The angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level.
- These angles are crucial in solving height and distance problems.
- If an observer is looking up at an object, the angle of elevation is formed between the line of sight and the horizontal.
- If an observer is looking down at an object, the angle of depression is formed between the line of sight and the horizontal.
- The angle of elevation from a point A to a point B is equal to the angle of depression from point B to point A.
Solving Height and Distance Problems
- Identify the right-angled triangle in the problem.
- Label the known and unknown quantities.
- Choose the appropriate trigonometric ratio (sin, cos, tan) based on the known and unknown sides and angles.
- Set up an equation and solve for the unknown quantity.
- Ensure that the units are consistent throughout the problem.
- Draw a clear diagram representing the problem, showing the angles of elevation or depression.
- Understand the relationship between the heights, distances, and angles.
Examples of Applications
- Finding the Height of a Tower: If the angle of elevation of the top of a tower from a point on the ground is known along with the distance from the base of the tower, one can calculate the height of the tower using the tangent function.
- Determining the Width of a River: By measuring the angle of elevation of a tree on the opposite bank from two different points on the same bank, one can calculate the width of the river using trigonometric ratios and properties.
- Navigation Calculations: Calculating the shortest distance between two points or determining the bearing required to reach a destination considering the wind or current effects.
- Surveying Land: Determining the area of a plot of land by dividing it into triangles and using trigonometric functions to calculate the lengths of sides and angles.
- Calculating the distance of a ship from a lighthouse, given the angle of depression from the top of the lighthouse to the ship.
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