Trigonometry

This paragraph introduces the audience to trigonometry, specifically for those studying it in high school or college. It explains the acronym SOHCAHTOA, which is a mnemonic for remembering the trigonometric ratios of sine, cosine, and tangent. The paragraph delves into how these ratios relate to the sides of a right triangle, specifically the opposite, adjacent, and hypotenuse with respect to an angle theta. An example using a 3-4-5 triangle is provided to illustrate the calculation of these ratios.

The paragraph continues the discussion on trigonometric ratios by showing how to apply them to find the sine, cosine, and tangent of an angle in a right triangle when two sides are given. It explains the use of the Pythagorean theorem to find the missing side of the triangle. The example triangle has sides of length 5 and 12, and the process to find the hypotenuse and subsequently the trigonometric ratios is demonstrated. The paragraph also touches on the concept of special right triangles and their memorization for quick calculations.

This section focuses on the use of special right triangles to simplify the evaluation of trigonometric ratios. It provides examples of such triangles, including the 3-4-5, 5-12-13, and others, emphasizing their utility in avoiding the use of the Pythagorean theorem. The paragraph also explains how to handle trigonometry problems where the ratios are given and the angle is between 0 and 90 degrees, using the example of a triangle with a sine ratio of 8/17.

The paragraph discusses the approach to finding trigonometric ratios when dealing with non-special triangles and angles that fall in different quadrants. It explains the use of the Pythagorean theorem to find missing sides and the importance of considering the quadrant to determine the signs of the ratios. An example is given where sine theta equals 2/5, and the angle is between pi/2 and pi, leading to the use of the Pythagorean theorem and consideration of the triangle's location in quadrant two.

This part of the script covers the process of rationalizing trigonometric expressions, particularly when the denominator contains a square root. It explains how to multiply the numerator and denominator by the square root to achieve a rationalized form. The paragraph also covers finding the cosecant, secant, and cotangent ratios for the given triangle, demonstrating the process with examples.

The paragraph explores how to find the exact values of trigonometric functions for specific angles without a calculator. It introduces the concept of special right triangles, such as the 30-60-90 triangle, and how to use them to evaluate functions like sine 30 and cosine 60. It also explains how to convert angles from radians to degrees and use the unit circle for such evaluations.

This section deals with evaluating trigonometric functions for angles greater than 90 degrees. It explains the concept of a reference angle, which is the acute angle formed between the terminal side of the angle and the x-axis. The paragraph demonstrates how to use the reference angle to find the values of sine, cosine, and tangent for angles in different quadrants, using examples like sine of 240 degrees and cosine of 150 degrees.

The paragraph discusses the signs of trigonometric functions in different quadrants using the mnemonic 'all students take calculus.' It explains which functions are positive in each quadrant and relates the signs to the coordinates of points within those quadrants. The concept of tangent as y/x and its implications for the signs of the functions is also covered, along with an example of evaluating tangent at negative 120 degrees.

This section covers the evaluation of cosecant and secant functions for angles in specific quadrants. It explains that secant is the reciprocal of cosine and cosecant is the reciprocal of sine. The paragraph provides examples of finding the secant of 225 degrees and cosecant of negative 13 pi/6, including the process of converting angles from radians to degrees and dealing with negative angles and coterminal angles.

The final paragraph wraps up the video by summarizing the content covered and encouraging viewers to check out additional trigonometry resources in the video description. It also prompts viewers to subscribe to the channel and turn on notifications to stay updated with new content.