Podcast
Questions and Answers
What does the 'S' in SOH CAH TOA represent?
What does the 'S' in SOH CAH TOA represent?
In a right triangle, which side is classified as the hypotenuse?
In a right triangle, which side is classified as the hypotenuse?
When using the Pythagorean theorem, which of the following equations correctly represents the relationship between the sides?
When using the Pythagorean theorem, which of the following equations correctly represents the relationship between the sides?
How do you find an unknown side when you know one side and one angle in a right triangle?
How do you find an unknown side when you know one side and one angle in a right triangle?
Signup and view all the answers
To find the angle in a triangle where the opposite side is 6 units and the adjacent side is 5 units, which function would be applicable?
To find the angle in a triangle where the opposite side is 6 units and the adjacent side is 5 units, which function would be applicable?
Signup and view all the answers
In the context of trigonometry, what information is needed to apply sine to find an angle?
In the context of trigonometry, what information is needed to apply sine to find an angle?
Signup and view all the answers
What is the approximate angle of elevation when a telephone pole 12 meters high is supported by a 14-meter wire?
What is the approximate angle of elevation when a telephone pole 12 meters high is supported by a 14-meter wire?
Signup and view all the answers
Which of the following is true regarding the adjacent side in relation to the reference angle?
Which of the following is true regarding the adjacent side in relation to the reference angle?
Signup and view all the answers
What is the total height of the tree after adding the height from the ground to the attachment point of the wires and the segment above it?
What is the total height of the tree after adding the height from the ground to the attachment point of the wires and the segment above it?
Signup and view all the answers
How is the length of the second wire determined?
How is the length of the second wire determined?
Signup and view all the answers
Which trigonometric function is used to find the horizontal distance from the base of the tree to the point where the first wire touches the ground?
Which trigonometric function is used to find the horizontal distance from the base of the tree to the point where the first wire touches the ground?
Signup and view all the answers
What is the length of the second wire calculated to be?
What is the length of the second wire calculated to be?
Signup and view all the answers
If one of the guy wires breaks and the new attachment point is 2 meters down from the top of the tree, what is the new height from the ground to this point?
If one of the guy wires breaks and the new attachment point is 2 meters down from the top of the tree, what is the new height from the ground to this point?
Signup and view all the answers
What angle does the replacement wire make with the ground after the old wire is replaced?
What angle does the replacement wire make with the ground after the old wire is replaced?
Signup and view all the answers
Which calculation is necessary to determine the total horizontal distance between the two wires?
Which calculation is necessary to determine the total horizontal distance between the two wires?
Signup and view all the answers
How do you calculate the height segment 'y' from the ground to the attachment point of the first wire?
How do you calculate the height segment 'y' from the ground to the attachment point of the first wire?
Signup and view all the answers
What is the significance of the 3-meter segment above the attachment points of the wires?
What is the significance of the 3-meter segment above the attachment points of the wires?
Signup and view all the answers
What does the calculation of 'a' and 'b' represent in the context of trigonometric applications for this scenario?
What does the calculation of 'a' and 'b' represent in the context of trigonometric applications for this scenario?
Signup and view all the answers
Study Notes
Right Triangle Trigonometry
- SOH CAH TOA is a helpful acronym for remembering trigonometric ratios:
- Sine (SOH): Opposite side / Hypotenuse
- Cosine (CAH): Adjacent side / Hypotenuse
- Tangent (TOA): Opposite side / Adjacent side
Identifying Sides in a Right Triangle
- The hypotenuse is always the longest side and is opposite the right angle.
- The opposite side is the side across from the reference angle.
- The adjacent side is the side next to the reference angle (not the hypotenuse).
- The reference angle is the angle used to determine the opposite and adjacent sides.
Solving for Unknown Sides
- Scenario 1: If two sides of a right triangle are known, use the Pythagorean theorem to find the third side.
- Pythagorean Theorem: c² = a² + b² (where c is the hypotenuse and a and b are the legs)
- Scenario 2: If one side and one angle are known, use a trigonometric ratio to find the unknown side.
- Choose the ratio that includes the known side and the unknown side.
Example: Finding the Length of a Side
- Example: A right triangle with a 49-degree angle, a hypotenuse of 17.6 units, and an unknown side opposite the angle.
- Use the sine ratio (since we know the hypotenuse and want the opposite side).
- sin(49°) = opposite / hypotenuse
- sin(49°) = x / 17.6
- Solve for x: x = 17.6 * sin(49°) ≈ 13.3 units
Finding an Angle
- Example: A right triangle with sides of 6 and 5 units, and we want to find the angle opposite the 6-unit side.
- Use the tangent ratio (relating opposite and adjacent sides).
- tan(x) = opposite / adjacent = 6 / 5
- Find the angle x using the inverse tangent function (tan⁻¹): x = tan⁻¹(6/5) ≈ 50.2°
Example: Finding the Angle of Elevation
- Example: A 12-meter high telephone pole is supported by a 14-meter wire. Find the angle between the wire and the ground (angle of elevation).
- Use the sine ratio (knowing opposite and hypotenuse).
- sin(θ) = opposite / hypotenuse = 12 / 14
- Find the angle θ using the inverse sine function (sin⁻¹): θ = sin⁻¹(12/14) ≈ 59°
Key Takeaways
- Trigonometric Ratios: Use SOH CAH TOA to remember relationships between sides and angles in right triangles.
- Pythagorean Theorem: Use this theorem to find a missing side when two sides are known.
- Inverse Trigonometric Functions: Use these functions to find an unknown angle when knowing the values of the trigonometric ratios.
- Application: Use trigonometry to solve practical problems involving right triangles (heights, distances, angles).
Trigonometry Applications: Real-World Problems
- Scenario: A large tree is moved, held upright by two guy wires.
- Details:
- One wire (12 meters) forms a 56-degree angle with the ground.
- The other wire makes a 48-degree angle with the ground.
- Both wires are attached 3 meters down from the top of the tree.
- Goal: Calculate the tree's total height, the second wire's length, and the wires' horizontal separation.
Finding the Height of the Tree
- Calculation: The tree's height splits into two parts: the segment from ground to wire attachment point ("y") and the 3-meter segment above.
- Using the sine function: sin(56°) = y / 12
- Solving for "y": y = 12 * sin(56°) ≈ 9.95 meters
- Total Height: y + 3 meters = 9.95 meters + 3 meters = 12.95 meters
Determining the Length of the Second Wire
- Calculation: The second wire's length ("r") is found using the triangle with the 48-degree angle and the height "y."
- sin(48°) = y / r
- Solving for "r": r = y / sin(48°) = 9.95 meters / sin(48°) ≈ 13.39 meters
Calculating the Horizontal Distance Between the Wires
- Splitting the Distance: The horizontal distance is the sum of two segments, "a" and "b."
- "a": From tree base to where the first wire touches the ground.
- "b": From tree base to where the second wire touches the ground.
- Calculating "a" and "b":
- Using the tangent function: tan(48°) = y / a => a = y / tan(48°) ≈ 8.96 meters
- Using the tangent function: tan(56°) = y / b => b = y / tan(56°) ≈ 6.71 meters
- Total Horizontal Distance: a + b = 8.96 meters + 6.71 meters ≈ 15.67 meters
Analyzing the Impact of a Broken Wire
- Scenario: One wire breaks; a replacement is needed.
- Change: The new wire is attached 2 meters below the tree top, giving a new height from the ground to attachment (10.95 meters).
- Goal: Determine the new angle the replacement wire makes with the ground.
Finding the New Angle
- Application of Sine: The sine function is used.
- sin(θ) = 10.95 meters / 13.39 meters
- Using the Inverse Sine: To find the angle (θ), use the inverse sine function (arcsin):
- θ = arcsin (10.95 meters / 13.39 meters) ≈ 54.9 degrees
- Rounding: Angle is approximately 55 degrees.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of right triangle trigonometry concepts, including the use of SOH CAH TOA, identifying sides, and solving for unknown sides. This quiz will cover key principles such as the Pythagorean theorem and trigonometric ratios. Sharpen your skills and see how well you know your triangles!
