Solving Trigonometric Equations

Front

What is the general solution for the trigonometric equation \(4\cot^2\theta = -1 - 2\cot\theta + 3\cot^2\theta\)?

Back

The solutions are \(\theta = \frac{3\pi}{4}, \frac{7\pi}{4}\) for \(0 \leq \theta < 2\pi\).

Front

What is the Pythagorean identity for sine and cosine?

Back

The Pythagorean identity states that \(\sin^2 x + \cos^2 x = 1\).

Front

What is the cotangent function in terms of sine and cosine?

Back

The cotangent function is defined as \(\cot x = \frac{\cos x}{\sin x}\).

Front

How do you solve a trigonometric equation involving cotangent?

Back

To solve a trigonometric equation involving cotangent, isolate \(\cot\theta\) and use the unit circle or reference angles to find solutions.

Front

What is the range of the cotangent function?

Back

The range of the cotangent function is all real numbers, \( (-\infty, \infty) \).

Front

What is the period of the cotangent function?

Back

The period of the cotangent function is \(\pi\).

Front

What is the relationship between the angles in a right triangle and the cotangent function?

Back

In a right triangle, \(\cot\theta = \frac{\text{adjacent}}{\text{opposite}}\).

Front

What is the value of \(\cot(0)\)?

Back

The value of \(\cot(0)\) is undefined because it involves division by zero.

Front

What is the complementary angle identity for cotangent?

Back

The complementary angle identity states that \(\cot(90^\circ - \theta) = \tan\theta\).

Front

How can you verify the solutions of a trigonometric equation?

Back

You can verify the solutions by substituting them back into the original equation to check if both sides are equal.