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00:00:00 – 00:24:01
The video focuses on solving trigonometric equations by combining trigonometry concepts with equation-solving techniques. The instructor demonstrates the process step by step, showcasing methods like using the unit circle, calculators, factoring, and inverse trigonometric functions. Key points include solving equations involving sine, tangent, and cosine functions with a restricted domain between 0 and 2π. The video emphasizes finding multiple solutions and adjusting them within the domain limits. The importance of not dividing by variables to prevent losing valid solutions is highlighted, along with the necessity of adding multiples of 2π for complete solutions. Concepts such as factoring, symmetrical angles, multiple solutions, and the repetitive nature of trigonometric values are discussed. The instructor provides guidance on visualizing solutions on the unit circle, graphing functions, and solving trig inequalities effectively.
00:00:00
In this segment of the video, the instructor focuses on combining trigonometry and solving equations. They begin by setting a domain limit between 0 and 2π. Exact values are obtained using the unit circle, while approximate values are calculated with a calculator. A trig equation 2sin(x) = -1 is solved step by step, isolating x by dividing and taking the inverse sine to find the solution as x = 7π/6 or 11π/6 within the restricted domain. The instructor emphasizes both manual and calculator-based approaches to solving trigonometric equations effectively.
00:03:00
In this segment of the video, the instructor is solving a trigonometric equation involving tangents. They factor the equation and set each term equal to zero to find solutions. The instructor uses the inverse tangent function to find the angle where tangent equals -4, but since the calculator only gives values in the first and fourth quadrants, the instructor reminds the viewer to consider additional solutions. The instructor then adjusts the solution to fit within the restricted domain by adding 2 pi.
00:06:00
In this segment of the video, the presenter demonstrates adding 2 pi to an angle and finding the exact decimal value using a calculator. They show two possible answers within the restricted domain by adding 2 pi and pi to two different angles. The presenter explains using inverse tangent to find another angle, and adds pi to reach the symmetrical angle on the other side. The segment concludes with a total of four possible answers within the restricted domain between zero and two pi.
00:09:00
In this part of the video, the speaker explains how to solve an equation involving cosine values. By square rooting both sides of the equation, the speaker simplifies it to cosine theta equals plus or minus the square root of two over two. This results in multiple solutions on the unit circle, specifically at pi over 4 and 3 pi over 4. The solutions can be expressed as pi over 4 plus pi n and 3 pi over 4 plus pi n, where n is an integer. Various possible angles can be determined by listing the solutions individually or using a general formula.
00:12:00
In this segment of the video, the speaker discusses solving a factoring problem involving trigonometric functions. The key point emphasized is not to divide by variables on both sides as it may lead to the cancellation of valid solutions. The speaker demonstrates the process of factoring by finding the greatest common factor, which in this case is sin(theta). They show the steps to solve for theta and explain the possible values for the solution, involving zero and pi. Additionally, the speaker uses a calculator to find the inverse sine of 1/3, obtaining an angle of approximately 33.9 degrees as a solution.
00:15:00
In this segment of the video, the speaker is discussing finding angles in the second quadrant. They show how to calculate the angle 339 degrees above the x-axis but just short of pi. The angle is determined to be approximately 2.8017 radians. The speaker emphasizes the need to add 2pi to these values due to the repetitive nature of approximate values. They also mention checking solutions by graphing trigonometric functions and solving equations step by step. The segment ends with guidance on solving an inequality involving the cinex function.
00:18:00
In this segment of the video, the instructor demonstrates how to find the inverse cosine of negative 1 within a restricted domain of 0 to 2pi. By analyzing the unit circle, it is determined that the values occur at 2pi over 3 and 4pi over 3. A rough sketch of the cosine function graph is made to visualize the solution points, which are identified as -2pi over 3 and 4pi over 3. The solution set for x values that satisfy the inequality lies between 2pi over 3 and 4pi over 3.
00:21:00
In this segment of the video, the instructor explains how to solve a trigonometric equation involving cosine of 2 theta equaling zero. By taking the inverse cosine of both sides, the equation simplifies to 2 theta. After dividing by two, the solution becomes theta equals pi over 4 plus pi over 2n. The instructor also highlights how adjusting the period of the cosine function can lead to multiple answers in the 2pi range. This demonstrates how trig inequalities can complicate solutions.