Gr12 Mathematics: Ch 4.3 Solving Equations

What is the first step in finding the general solution of a trigonometric equation?

Determine the reference angle using positive values.

Determine where the function is positive or negative using the CAST diagram.

Find angles within a specified interval by adding or subtracting multiples of the period.

Simplify the equation using algebraic methods and trigonometric identities. (correct)

If (\sin \theta = x), what are the general solutions?

\(\theta = \sin^{-1} x + k \cdot 180^\circ\) or \(\theta = 360^\circ - \sin^{-1} x + k \cdot 180^\circ\)

\(\theta = \cos^{-1} x + k \cdot 360^\circ\) or \(\theta = 360^\circ - \cos^{-1} x + k \cdot 360^\circ\)

\(\theta = \tan^{-1} x + k \cdot 180^\circ\)

\(\theta = \sin^{-1} x + k \cdot 360^\circ\) or \(\theta = 180^\circ - \sin^{-1} x + k \cdot 360^\circ\) (correct)

What is the period used in finding the general solution for (\tan \theta = x)?

360

180 (correct)

45

90

What is the purpose of using the CAST diagram in finding the general solution of a trigonometric equation?

To determine where the function is positive or negative. Signup and view all the answers

What is the last step in finding the general solution of a trigonometric equation?

Verify the solutions using a calculator. Signup and view all the answers

What is the purpose of determining the reference angle in finding the general solution of a trigonometric equation?

To determine where the function is positive or negative. Signup and view all the answers

What is the general solution for (\cos \theta = x)?

(\theta = \cos^{-1} x + k \cdot 360^\circ) Signup and view all the answers

Why do we need to add or subtract multiples of the period in finding the general solution?

To find angles within a specified interval. Signup and view all the answers

What is the purpose of checking the solutions using a calculator in finding the general solution?

To verify the solutions. Signup and view all the answers

What is the general solution for (\tan \theta = x)?

(\theta = \tan^{-1} x + k \cdot 180^\circ) Signup and view all the answers

Why do we need to consider the period of the trigonometric function in finding the general solution?

To find infinitely many angles that satisfy the equation. Signup and view all the answers

What is the main reason for the existence of infinitely many angles that satisfy a given trigonometric equation?

The periodic nature of trigonometric functions Signup and view all the answers

If (\sin \theta = x) and (k) is an integer, what is the general solution for (\theta)?

(\theta = \sin^{-1} x + k \cdot 360^\circ) Signup and view all the answers

What is the purpose of the CAST diagram in solving trigonometric equations?

To determine where the function is positive or negative Signup and view all the answers

If (\cos \theta = x), what is the general solution for (\theta) if (k) is an integer?

(\theta = \cos^{-1} x + k \cdot 360^\circ) Signup and view all the answers

What is the result of adding or subtracting multiples of the period in finding the general solution?

We get infinitely many angles that satisfy the equation Signup and view all the answers

What is the relationship between the period of the function and the general solution?

The period is used to find infinitely many angles that satisfy the equation Signup and view all the answers