How To Solve Two Triangle Trigonometry Problems
TLDRThis educational script teaches how to solve for unknown sides in right triangle systems using trigonometric functions. It demonstrates step-by-step methods to find the value of 'x' in various scenarios by breaking down complex problems into simpler right triangles. The script utilizes SOHCAHTOA mnemonic to apply sine, cosine, and tangent functions based on given angles and sides, solving for 'h1', 'h2', and 'y' to eventually determine 'x'. It covers multiple examples, illustrating the process of setting up equations and using tangent values to find unknown lengths, providing a clear guide for students to understand and apply trigonometry in geometric problems.
Takeaways
- 📚 The script discusses methods for solving right triangles, especially when dealing with a system of two right triangles.
- 📐 It introduces a problem involving two right triangles with given angles and a distance, aiming to find the value of x, which is the sum of two segments (h1 and h2).
- 🔍 The script uses trigonometric functions, specifically the tangent function, to solve for the unknown lengths in the triangles.
- 📈 The first example calculates h1 and h2 separately using the tangent of the given angles (40° and 30°) and then sums them to find x.
- 📝 In the second example, the script shows how to find the value of x by subtracting h1 from h2, after calculating both using the tangent of 30° and 39° respectively.
- 🧩 The third example involves breaking down a larger problem into two smaller triangles and using the tangent function to find the lengths h1 and h2, then finding x by subtraction.
- 📉 The fourth example is a more complex problem where the difference between two segments (h2 and h1) is given as 80, and the script sets up a system of equations to solve for x and y.
- 🔢 The script emphasizes the importance of using decimal values for trigonometric calculations and provides the actual numerical solutions for the tangent of 43° and 35°.
- 🔍 The final part of the script solves for y by setting up an equation from the tangent values and the given distance, and then uses this value to find x.
- 📝 The process involves algebraic manipulation, such as multiplying both sides of an equation by a common factor and canceling terms to isolate the variable.
- 🎯 The script concludes by finding the value of x using the calculated value of y and the tangent of the given angle, demonstrating a step-by-step approach to problem-solving in trigonometry.
Q & A
What is the purpose of the video script?
-The video script is aimed at explaining how to solve certain right triangles, particularly when dealing with a system of two right triangles, using trigonometric functions and the SOHCAHTOA mnemonic.
What is the first step in solving the given problem with two right triangles?
-The first step is to separate the system into two smaller right triangles and then apply trigonometric functions to find the unknown side lengths.
Which trigonometric function is used to find the value of h1 when given an angle of 40 degrees and a distance of 800?
-The tangent function is used because h1 is opposite to the 40-degree angle and 800 is adjacent to it. The formula used is tan(40) = h1 / 800.
What is the value of h1 when the angle is 40 degrees and the adjacent side is 800?
-The value of h1 is calculated as 800 times the tangent of 40 degrees, which is approximately 671.3.
How is the value of h2 found when dealing with a 30-degree angle and an adjacent side of 800?
-The value of h2 is found using the tangent function, where tan(30) = h2 / 800, and then multiplying both sides by 800 to get h2, which is approximately 461.9.
What is the sum of h1 and h2 in the first problem, and what is its significance?
-The sum of h1 and h2 is 1133.2. This value represents the total length of the side x in the original problem.
In the second problem, how is the value of x related to h1 and h2?
-In the second problem, x is the difference between h2 and h1, as h2 is the sum of x and h1.
What is the tangent of 30 degrees and how is it used to find h1 in the second problem?
-The tangent of 30 degrees is approximately 0.57735. It is used to find h1 by multiplying 200 (the hypotenuse) by the tangent of 30 degrees to get h1, which is about 115.5.
In the third problem, how are the values of h1 and h2 related to the value of x?
-In the third problem, x is the difference between h2 and h1. After finding h1 and h2 separately, they are subtracted to get the value of x.
How is the value of x calculated in the fourth problem with angles of 43 and 35 degrees and a distance of 80?
-In the fourth problem, a system of equations is set up using the tangent function for both angles. After finding the value of y, x is calculated using the equation x = y * tan(43), which gives a value of approximately 224.6.
Outlines
📐 Solving Right Triangles with Two Adjacent Triangles
This paragraph introduces a method to solve for the unknown side lengths in a system of two right triangles. Given the angles and a side length, the process involves using trigonometric functions, specifically the tangent function, to find the lengths of the other sides. The example demonstrates finding the value of 'x' by breaking down the problem into two smaller right triangles and applying the tangent function to find the lengths of 'h1' and 'h2', which are then summed to get 'x'. The calculations include multiplying the given side length by the tangent of the given angles to find the opposite sides and then summing these to find the total distance 'x'.
🔍 Calculating Distances Using Trigonometry
The second paragraph continues the theme of using trigonometry to solve right triangles but introduces a different scenario where the goal is to find the distance 'x' between two points given the angles and a base length. The method involves breaking the problem into two right triangles and using the tangent function to find the lengths of 'h1' and 'h2'. The process is explained step by step, with the calculation of 'h1' using the tangent of 30 degrees and the calculation of 'h2' using the tangent of 39 degrees. The final step is to find 'x' by subtracting 'h1' from 'h2', yielding the distance between the two points.
🧩 Solving for 'x' in a Complex Triangle Scenario
This paragraph presents a more complex problem involving two angles and a known distance between a section of the triangle. The challenge is to find the value of 'x', which represents the length of one side. The approach is to introduce a new variable 'y' and set up a system of equations based on the tangent function for the two triangles formed by the angles and the known distance. The solution involves finding the tangent values for the given angles, setting up an equation that equates 'y' times the tangent of one angle to the sum of the known distance and 'y' times the tangent of the other angle, and solving for 'y'. Once 'y' is found, 'x' can be calculated using the relationship established in the equations.
🎯 Final Calculation for 'x' in a Given Triangle Setup
The final paragraph wraps up the series of problems by presenting the solution for finding 'x' in the last triangle scenario. It reiterates the process of using the tangent function to solve for 'x' by relating it to the variable 'y'. The paragraph concludes with the calculation of 'x' using the value of 'y' and the tangent of the given angle, providing the final answer to the problem. The summary emphasizes the methodical approach to solving such trigonometric problems and the importance of setting up and solving a system of equations when dealing with multiple unknowns.
Mindmap
Keywords
💡Right Triangles
💡SOHCAHTOA
💡Tangent Function
💡Opposite Side
💡Adjacent Side
💡Hypotenuse
💡Angle
💡System of Equations
💡Trigonometric Ratios
💡Distance
Highlights
Introduction to solving right triangles in a two-triangle system.
Use of angle measures 40 and 30 degrees to find the value of x in a right triangle.
Application of SOHCAHTOA to determine the value of h1 using the tangent function.
Calculation of h1 as 800 times the tangent of 40 degrees.
Finding the value of h2 using a 30-degree angle and the tangent function.
Summation of h1 and h2 to determine the value of x in the triangle system.
Method to solve for x when given two angles and a distance in a triangle.
Utilization of tangent function to find h1 in a 30-degree triangle.
Calculation of h1 as 200 times the tangent of 30 degrees.
Determination of h2 using a 39-degree angle and the tangent function.
Finding the value of x as the difference between h2 and h1.
Approach to solve for x with given angles of 25 and 35 degrees and a side length.
Use of tangent function to calculate h1 in a 35-degree triangle.
Calculation of h2 using a 25-degree angle and the tangent function.
Finding the value of x as the difference between h2 and h1 in another triangle scenario.
Introduction of a more complex problem with angles 43 and 35 degrees and a distance of 80.
Strategy to find x by setting up a system of equations with two variables.
Solution for y using the tangent of 43 degrees and the tangent of 35 degrees.
Final calculation of x using the value of y and the tangent function.
Transcripts
5.0 / 5 (0 votes)
Thanks for rating: