Solve each equation for exact solutions over the interval 2...

Key Concepts

Exact Trigonometric Values

In many trigonometric problems, especially those seeking exact solutions, it is important to use known values from the unit circle. These exact values correspond to specific angles where sine, cosine, and other trigonometric functions have well-defined values. Recognizing these can simplify solving equations and ensure the solutions are expressed in exact terms, often in relation to ?.

Periodicity of Trigonometric Functions

Trigonometric functions are periodic, meaning their values repeat after a fixed interval. For the cotangent function, the period is ?. This property is critical when solving equations because it ensures that once a solution is found, an infinite set of solutions exists, differing by integer multiples of the period. When restrictions such as specific intervals are applied, only the solutions within that range are considered.

Cotangent Function

The cotangent function is one of the six primary trigonometric functions, defined as the ratio of the cosine to the sine of an angle. It is used in various trigonometric equations and has unique properties, such as being undefined when the sine is zero, and it has a period of ?. Understanding cotangent is essential for solving equations where it appears, as it can often be related to other trigonometric functions like tangent.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and then determining all the angles that satisfy the resulting equation within the given interval. This process often includes algebraic manipulation to simplify the equation and applying inverse trigonometric functions or known unit circle values to obtain exact solutions.

Transcript

00:01 We are asked to solve the equation 2 times the cotangent of x plus 1 equals negative 1.

00:08 Certainly, our first goal here is going to be to try and isolate that co -tangent of x, and we'd start by subtracting 1 to both sides of the equation.

00:19 So that would give us 2 times the co -tangent of x equals negative 2.

00:26 Then we'd want to divide by 2, and that would give us the co -tangent of x equals negative 2.

00:32 X is equal to negative 1.

00:38 We know that cotangent is the inverse, the reciprocal, of tangent.

00:44 So cotangent of x, we could rewrite as 1 over tangent of x, and that is still equal then to negative 1.

00:57 If 1 over the tangent of x is equal to negative 1, then that means the tangent of x is simply going to be equal to the reciprocal of negative 1.

01:08 Or, to show that algebraically, we would want to multiply by the tangent of x on both sides of the equation.

01:18 That's going to cancel out there.

01:20 So we'd have 1 equals the negative tangent of x.

01:27 To get tangent of x by itself, we would need to divide by 1, negative 1, i mean, and that's going to give us that the tangent of x is equal to negative 1.

01:43 That's a long way of doing it, but just wanted to show that in case you needed to show them.

01:48 So, where we're at right now is we have the tangent of x is equal to negative 1.

01:55 We want to go to our unit circle to solve this out...

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