Solving Trigonometric Equations I

Mathispower4u

4 Jun 201008:19

EducationalLearning

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TLDRThis instructional video introduces viewers to solving basic trigonometric equations in linear form with a single trigonometric function. The presenter demonstrates how to find solutions for sine, cosine, and tangent equations on the interval from 0 to 2π and then extends the solutions to all radian measures. Techniques such as using the unit circle and understanding trigonometric identities are highlighted. The video also covers how to handle equations with no solution due to the range restrictions of trigonometric functions. It is the first in a series of videos aimed at mastering trigonometric equation solving.

Takeaways

📚 This video is the first in a series focused on solving trigonometric equations.
🔍 The goal is to solve basic trig equations, similar to solving algebraic equations, using various techniques.
📉 The video specifically addresses solving trig equations in linear form with a single trigonometric function.
🔑 The initial approach is to solve for the interval from 0 to 2π and then extend the solutions to all radian measures.
⚡ To solve for sine theta, the equation is manipulated algebraically to isolate sine theta and set it equal to 1/2.
📐 The unit circle is used to find angles with a sine value of 1/2, focusing on the first and second quadrants.
📈 The angles found are 30 degrees (π/6 radians) and 150 degrees (5π/6 radians) within the 0 to 2π interval.
🔄 The general solution for sine theta involves adding multiples of 2π to the found angles to account for the sine function's periodicity.
📉 For cosine theta, the process involves finding x-coordinates on the unit circle that correspond to the given value.
📊 The video demonstrates that there are no solutions when the value of cosine theta exceeds its possible range of -1 to 1.
📐 The tangent function's solutions are found by considering the positive values of tangent in the first and third quadrants, using the 30-60-90 triangle ratio.
🔄 The general solution for tangent theta includes adding multiples of π to the base angle due to the tangent function's periodicity of π.

Q & A

What is the primary goal of the video series?
-The primary goal of the video series is to teach viewers how to solve trigonometric equations using various methods.
What is the focus of the first video in the series?
-The first video focuses on solving the most basic type of trigonometric equations, specifically those in linear form with one trigonometric function.
What is the first step in solving a trigonometric equation in linear form?
-The first step is to isolate the trigonometric function, such as sine or cosine, on one side of the equation.
Why is it necessary to recognize which technique to use when solving trigonometric equations?
-Different trigonometric equations require different solving techniques, and recognizing the appropriate method is crucial for finding the correct solution.
What is the interval range for the initial solution of trigonometric equations in the video?
-The initial solutions are found on the interval from zero to 2π radians.
How does the video demonstrate finding the angles with a sine function value of 1/2?
-The video uses the unit circle, noting that the y-coordinate on the unit circle is equal to sine theta, and identifies the angles in the first and second quadrants that have a y-coordinate of 1/2.
What is the significance of the unit circle in solving trigonometric equations?
-The unit circle is significant because it helps in identifying angles that correspond to specific trigonometric function values, such as sine or cosine.
How does the video explain finding all radian solutions for a trigonometric equation?
-The video explains that to find all radian solutions, one must consider the period of the trigonometric function and add multiples of this period to the initial solutions.
What is the period of the sine function, and how does it relate to finding all radian solutions?
-The period of the sine function is 2π radians. To find all radian solutions, one can add any multiple of 2π radians to the initial solutions.
How does the video handle the case where the cosine function value is set to 2?
-The video explains that since the range for cosine theta is between -1 and +1, there cannot be a solution where cosine theta equals 2.
What is the graphical representation used in the video to illustrate the solutions of trigonometric equations?
-The video uses graphs of y = sine theta, y = cosine theta, and y = tangent theta, along with horizontal lines at the specific function values to show the intersection points, which represent the solutions.
Why is the tangent function value of 1/√3 significant in the video?
-The tangent function value of 1/√3 is significant because it corresponds to the tangent ratio in a 30-60-90 right triangle, which helps in identifying the angles that are solutions to the equation.
What is the period of the tangent function, and how does it affect the solution of trigonometric equations involving tangent?
-The period of the tangent function is π radians. This means that to find all radian solutions involving tangent, one can add multiples of π to the initial solutions.

Outlines

00:00

📚 Introduction to Solving Trigonometric Equations

This paragraph introduces a series of videos dedicated to solving trigonometric equations. The focus is on the most basic type of trigonometric equations, emphasizing that various methods exist for solving them, similar to algebraic equations. The video aims to teach how to solve linear trig equations involving a single trigonometric function, with an initial focus on the interval from zero to 2π radians. The process involves solving for sine theta, finding angles with a sine value of 1/2 within the specified interval, using the unit circle for reference. The solution set is then extended to all radian measures by considering the periodicity of the sine function, leading to a general solution involving an integer variable k. The paragraph concludes with a graphical representation of the sine function and its intersections with y = 1/2, illustrating the solution points.

05:05

📐 Solving Equations with Sine, Cosine, and Tangent

The second paragraph delves into solving trigonometric equations involving sine, cosine, and tangent functions. It begins with an equation for cosine theta, which is manipulated algebraically to isolate cosine theta and find its value as -√2/2. Using the unit circle, the solution identifies the angles in the second and third quadrants that correspond to this cosine value. The periodic nature of the cosine function is then discussed, with solutions extended to all radian measures by adding multiples of 2π. The paragraph also addresses an equation involving tangent theta, which is simplified to find a tangent value of 1/√3, reminiscent of a 30-60-90 triangle. The solutions for theta in the first and third quadrants are identified, and the periodicity of the tangent function is used to generalize the solution. The video script also highlights an equation with no solution due to the range of the cosine function, graphically illustrating the absence of intersection between y = 2 and y = cosine theta. The paragraph concludes with a reminder that this video is the first in a series on solving trigonometric equations.

Mindmap

Keywords

💡Trig Equations

Trig equations refer to mathematical equations that involve trigonometric functions such as sine, cosine, and tangent. In the context of the video, the theme revolves around solving these equations using various techniques. The script explains how to approach trig equations in linear form with a single trigonometric function, which is fundamental to understanding more complex trigonometric problems.

💡Solving Techniques

Solving techniques are the methods used to find the solutions to mathematical problems. The video emphasizes the importance of recognizing which technique to apply when tackling trig equations. It discusses several methods, indicating the necessity of practice to become proficient in solving trig equations effectively.

💡Linear Form

Linear form in trigonometry refers to equations where the trigonometric function is to the first power. The video specifically addresses how to solve trig equations in this form, which simplifies the process as compared to equations with higher powers or products of trigonometric functions.

💡Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. It is a fundamental tool in trigonometry for understanding and solving trig equations. In the script, the unit circle is used to determine angles that have specific sine and cosine values, such as 1/2 or -√2/2, by examining the coordinates on the circle.

💡Reference Triangles

Reference triangles are right-angled triangles with known side ratios that are used to remember the values of trigonometric functions for specific angles. Although not used extensively in the script, they are mentioned as an alternative to the unit circle for finding angles with sine values of 1/2.

💡Sine Function

The sine function is one of the primary trigonometric functions that relates the ratio of the opposite side to the hypotenuse in a right-angled triangle. In the video, the sine function is used to find angles that have a sine value of 1/2, with examples given for angles in the first and second quadrants.

💡Cosine Function

The cosine function is another fundamental trigonometric function, representing the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The script uses the cosine function to find angles with a specific cosine value, such as -√2/2, by examining the x-coordinates on the unit circle.

💡Tangent Function

The tangent function, denoted as tan, is the ratio of the opposite side to the adjacent side in a right-angled triangle. It is used in the video to solve for angles where the tangent value is given, such as 1/√3, by relating it to the angles in a 30-60-90 triangle.

💡Periodicity

Periodicity refers to the property of a function that repeats its values at regular intervals or periods. The sine and cosine functions have a period of 2π radians, while the tangent function has a period of π radians. The script explains how to find all solutions to trig equations by adding multiples of these periods to the initial solutions found within the interval from 0 to 2π.

💡No Solution

A 'no solution' scenario occurs when an equation does not have any values that satisfy the equation within a given domain. In the script, an example is given where cosine theta cannot equal 2 because the range of the cosine function is limited to the interval [-1, 1]. This illustrates the importance of understanding the range of trigonometric functions when solving equations.

Highlights

Introduction to a series of videos on solving trigonometric equations.

The goal is to solve basic trig equations using several methods.

Explanation of how to solve trig equations in linear form with one trig function.

Solving equations initially on the interval from 0 to 2π, then over all radian measures.

Demonstration of solving for sine theta by simplifying the equation.

Using the unit circle to find angles with a sine function value of 1/2.

Identification of two angles within the interval 0 to 2π that satisfy the sine equation.

Explanation of how to find all radian solutions by adding multiples of 2π.

Graphical representation of the sine function and its solutions.

Solving for cosine theta using the unit circle and identifying angles in specific quadrants.

Graphical illustration of the cosine function and its solutions.

Solving for tangent theta by relating it to a 30-60-90 triangle.

Graphical analysis of the tangent function and its solutions.

Explanation of the period for tangent theta being π radians.

Solving an equation with no solution due to the range of cosine theta.

Graphical verification of no solution for an equation with cosine theta.

Conclusion and encouragement to watch further videos on solving trig equations.

Transcripts

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Takeaways

Q & A

What is the primary goal of the video series?

What is the focus of the first video in the series?

What is the first step in solving a trigonometric equation in linear form?

Why is it necessary to recognize which technique to use when solving trigonometric equations?

What is the interval range for the initial solution of trigonometric equations in the video?

How does the video demonstrate finding the angles with a sine function value of 1/2?

What is the significance of the unit circle in solving trigonometric equations?

How does the video explain finding all radian solutions for a trigonometric equation?

What is the period of the sine function, and how does it relate to finding all radian solutions?

How does the video handle the case where the cosine function value is set to 2?

What is the graphical representation used in the video to illustrate the solutions of trigonometric equations?

Why is the tangent function value of 1/√3 significant in the video?

What is the period of the tangent function, and how does it affect the solution of trigonometric equations involving tangent?