11 - Learn ArcSin, ArcCos & ArcTan (Inverse Sin, Cos & Tan) - Part 1

The lesson begins with an introduction to inverse trigonometric functions, specifically arc sine, arc cosine, and arc tangent. The instructor expresses excitement about teaching these concepts, acknowledging their complexity and the common confusion they cause in traditional textbooks. The goal is to provide a clear understanding of these functions, emphasizing their importance in various fields such as algebra, trigonometry, calculus, physics, and engineering. The lesson aims to clarify the behavior of these functions and address common misconceptions by walking through detailed problem sequences.

This paragraph delves into the basics of inverse trigonometric functions, explaining the concept of the inverse operation. It highlights how the arc sine, arc cosine, and arc tangent functions are the inverses of their respective trigonometric functions, allowing for the reverse engineering of angles from their sine, cosine, or tangent values. The instructor clarifies the input and output ranges for these functions, noting that while the input values are limited (between -1 and 1 for sine and cosine, and between negative infinity and positive infinity for tangent), the output is a specific range of angles (between -90 to 90 degrees or -π/2 to π/2 radians).

The instructor addresses the challenges of understanding inverse trigonometric functions, particularly the confusion that arises from the multiple angles that can produce the same sine, cosine, or tangent value. This is due to the periodic nature of these functions and the existence of multiple solutions within the unit circle. The lesson emphasizes the importance of understanding the logic behind these functions and the limitations of calculators and computers when providing solutions, which are constrained to specific ranges of angles.

The paragraph provides examples to illustrate how inverse trigonometric functions work. It explains how the arc sine of a value gives an angle whose sine is that value, and similarly for arc cosine and arc tangent. The examples use known values such as the sine of 30 degrees (π/6 radians) and the cosine of 30 degrees to demonstrate the process of finding the corresponding angles. The paragraph also introduces the notation for inverse functions, clarifying that the negative exponent does not indicate raising the function to a negative power but rather denotes the inverse operation.

The instructor uses the analogy of a function as a box to explain the process of inputting numbers into inverse trigonometric functions and getting angles as outputs. It emphasizes the importance of remembering that angles, not just numbers, are the outputs when working with these functions. The paragraph also introduces the concept of a 'gotcha' related to the multiple angles that can satisfy a given trigonometric equation, highlighting the need for a thorough understanding of the underlying principles to avoid confusion.

This section discusses the process of solving equations using inverse trigonometric functions. It explains how to isolate the variable by applying the inverse function to both sides of the equation, effectively 'undoing' the trigonometric operation. The example of solving for an angle when given the sine value of 1/2 is used to illustrate this process. The paragraph also introduces the concept of the unit circle and how it relates to finding the correct angle among the infinite possibilities that satisfy the given trigonometric value.

The paragraph explores the concept of multiple angles around the unit circle that can produce the same trigonometric value. It uses the sine function as an example, showing that while the sine of π/6 (30 degrees) is 1/2, other angles such as 5π/6 and 13π/6 also have a sine value of 1/2. This highlights the challenge of determining the correct angle when using inverse trigonometric functions, as there are infinitely many angles that can satisfy a given sine, cosine, or tangent value. The paragraph sets the stage for a deeper discussion on how to handle these multiple solutions in trigonometric equations.

The lesson continues to clarify the range of inputs and outputs for inverse trigonometric functions. It explains that while the sine function can only output values between -1 and 1, the arc sine function can only accept inputs in the same range and will output angles between -π/2 and π/2. Similarly, the arc cosine function accepts inputs between -1 and 1 and outputs angles between 0 and π. The tangent function, which can output values ranging from negative infinity to positive infinity, accepts inputs in this range as well, but the arc tangent function outputs angles only between -π/2 and π/2, covering all possibilities for the tangent function's range.

The paragraph summarizes the key points about inverse trigonometric functions, emphasizing the importance of understanding the range of angles they can output. It reiterates that the calculator will provide the smallest possible angle that satisfies the input value, which is the fundamental angle. The paragraph also reminds that while there are multiple angles around the unit circle that can produce the same trigonometric value, the calculator will only return the fundamental angle, and additional angles can be found by adding or subtracting multiples of 180 degrees or 360 degrees to move into other quadrants.