# Evaluate Inverse Trig Functions - Step by Step

TLDRThis video tutorial explains the concept and application of inverse trigonometric functions, emphasizing the importance of understanding their origins and purpose. It covers the restrictions placed on these functions, such as the domain for sine (-π/2 to π/2), cosine (0 to π), and tangent (-π/2 to π/2), to ensure they are one-to-one and invertible. The video uses four examples to illustrate how to evaluate these functions, highlighting the need to find angles that satisfy the given restrictions and the process of using the unit circle to determine these angles.

###### Takeaways

- 📚 Understanding inverse trigonometric functions involves knowing their origins and purposes.
- 🔍 Inverse trigonometric functions 'undo' the regular trigonometric functions.
- 🤔 The sine of an angle (θ) can be thought of as the y-coordinate of a point on the unit circle.
- 📊 Evaluating trigonometric functions requires recognizing that they may have multiple solutions, but inverse functions need a single solution.
- 🚫 To find the inverse, restrict the domain of the sine function to between -π/2 and π/2.
- 📐 For cosine, the domain is restricted to between 0 and π.
- 🔢 The tangent function is a ratio of y to x, which complicates the process of finding the inverse.
- 📍 Tangent is negative in the second and fourth quadrants, which helps in determining the correct angle for the inverse.
- 🌐 The restrictions ensure that the inverse trigonometric functions are one-to-one, making them invertible.
- 📌 The restrictions for tangent are the same as for sine, from -π/2 to π/2.
- ⏩ Secant is often confused with the inverse or reciprocal, but it can be rewritten in terms of cosine for easier evaluation.

###### Q & A

### What are inverse trigonometric functions used for?

-Inverse trigonometric functions are used to 'undo' the results of the standard trigonometric functions, allowing us to find the angle when we know the value of a trigonometric function for that angle.

### Why do we need to understand where the inverse trigonometric functions come from?

-Understanding the origin of inverse trigonometric functions helps us grasp the concept of how they can reverse the process of standard trigonometric functions, which is crucial for their application in various mathematical and real-world problems.

### What is the restriction for the sine function when evaluating its inverse?

-When evaluating the inverse sine function, the angle (theta) must be between -π/2 and π/2.

### How many solutions does the equation sin(θ) = √3/2 have within the given restriction?

-Within the given restriction, there is one solution: θ = π/3.

### What is the restriction for the cosine function when evaluating its inverse?

-When evaluating the inverse cosine function, the angle (theta) must be between 0 and π.

### What angle satisfies the equation cos(θ) = -√2/2 within the given restriction?

-Within the given restriction, the angle that satisfies the equation is θ = 3π/4.

### Why are there two potential solutions for the equation tan(θ) = -√3/2?

-There are two potential solutions because tangent is a periodic function and can have two angles in different quadrants that yield the same value. However, when finding the inverse, we must consider the restriction to select the valid solution.

### What are the restrictions for the tangent function when evaluating its inverse?

-When evaluating the inverse tangent function, the angle (theta) must be between -π/2 and π/2.

### How can we find the angle that satisfies the equation cot(θ) = -√3/2?

-By rewriting the equation as the cosine of theta equals 1 over √(1 - (√3/2)^2), we find that the angle must be such that cos(θ) = 0, which is at θ = 0 within the given restriction.

### Why do we have restrictions on the angles in inverse trigonometric functions?

-Restrictions are necessary to ensure that the inverse trigonometric functions are one-to-one, meaning each output has a unique input. Without restrictions, the functions would fail the horizontal line test and not be invertible.

### What is the primary reason for the different restrictions on sine, cosine, and tangent functions?

-The different restrictions arise because of the periodic nature of these functions and their behavior in different quadrants of the unit circle. The restrictions ensure that the inverse functions are valid and can be accurately determined.

###### Outlines

##### 📚 Understanding Inverse Trigonometric Functions

This paragraph introduces the concept of inverse trigonometric functions and explains their purpose as the 'undo' operation for trigonometric functions. It emphasizes the importance of understanding their origins and the reasons for their existence. The speaker outlines a plan to cover four different examples to clarify these functions, their restrictions, and their applications. The first example uses the sine function and its inverse to illustrate how to find an angle whose sine value is equal to a given number, highlighting the concept of coterminal angles and the need for restricting the domain of the sine function to between -π/2 and π/2 to ensure one-to-one correspondence.

##### 📐 Evaluating Inverse Trigonometric Functions with Restrictions

The second paragraph delves into the specifics of evaluating inverse trigonometric functions, focusing on the restrictions that must be applied to ensure the functions are one-to-one. It covers the inverse cosine function, explaining how to find the angle that satisfies a given cosine value within the restricted domain of [0, π]. The discussion then moves to the tangent function, highlighting the challenges of dealing with y/x form and the quadrant restrictions for tangent. The paragraph also touches on the cotangent function and its relationship with the sine function, emphasizing the importance of staying within the defined restrictions. Finally, the secant function is introduced as the inverse of the cosine function, with a focus on its evaluation and the application of restrictions.

###### Mindmap

###### Keywords

##### 💡Inverse Trigonometric Functions

##### 💡Unit Circle

##### 💡Restrictions

##### 💡Sine Function

##### 💡Cosine Function

##### 💡Tangent Function

##### 💡Cotangent Function

##### 💡Secant Function

##### 💡One-to-One Functions

##### 💡Horizontal Line Test

###### Highlights

The video aims to explain how to evaluate inverse trigonometric functions.

Understanding the origin and purpose of inverse trigonometric functions is crucial.

Four different examples are provided to illustrate the concept.

Inverse trigonometric functions are used to 'undo' trigonometric functions.

The sine function is used to find the angle when given a y-coordinate on the unit circle.

The sine of an angle equals the y-coordinate of the corresponding point on the unit circle.

The example of sine of theta equals square root of 3 over 2 is used to demonstrate the process.

The concept of coterminal angles and their infinite nature is discussed.

Restrictions on the sine function are explained to find the correct angle within the range of negative pi to pi.

The cosine function is explored with the example of cosine of what angle equals a negative value.

The cosine function is restricted between 0 and pi for inverse evaluation.

The tangent function's y over x ratio is discussed with an example.

Tangent is negative in the second and fourth quadrants, and the restrictions for the tangent function are explained.

The cotangent function is related to the tangent function, and its restrictions are aligned with the sine function.

The secant function is the inverse of the cosine function, and it is explained with a specific example.

The restrictions for sine, cosine, and tangent functions are summarized for clarity.

The importance of adhering to the restrictions to ensure the functions are one-to-one is emphasized.

The video provides a comprehensive review of inverse trigonometric functions and their application.

###### Transcripts

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