Grade 11 Trigonometric Equations RevisionExercise 5: (Answers On Page 350)1) Solve For $\theta$ In The Following Equations, Correct To 1 Decimal Place Where Applicable:a) $2 \sin \theta \tan \theta - \tan \theta = 0$ If $\theta

Grade 11 Trigonometric Equations Revision Exercise 5: Solving Trigonometric Equations

Introduction to Trigonometric Equations

Trigonometric equations are an essential part of mathematics, particularly in the field of trigonometry. These equations involve trigonometric functions such as sine, cosine, and tangent, and are used to solve problems in various fields like physics, engineering, and navigation. In this article, we will focus on solving trigonometric equations, specifically the ones involving sine, cosine, and tangent functions.

Solving Trigonometric Equations

To solve trigonometric equations, we need to use various techniques such as factoring, using trigonometric identities, and applying algebraic manipulations. In this exercise, we will use these techniques to solve the given trigonometric equations.

Exercise 5: Solving Trigonometric Equations

1. Solve for $\theta$ in the following equations, correct to 1 decimal place where applicable:

a) $2 \sin \theta \tan \theta - \tan \theta = 0$

To solve this equation, we can start by factoring out the common term $\tan \theta$ from the left-hand side of the equation.

import math

# Define the equation
def equation(theta):
    return 2 * math.sin(theta) * math.tan(theta) - math.tan(theta)

# Solve for theta
theta = math.atan(1/2)  # Using the arctangent function to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the math.atan function to find the value of $\theta$ that satisfies the equation. The math.atan function returns the arctangent of a given number, which is the angle whose tangent is that number. In this case, we use math.atan(1/2) to find the value of $\theta$ that satisfies the equation.

Solution

The value of $\theta$ that satisfies the equation is $\theta = \boxed{26.6}$ degrees.

2) $2 \cos \theta \sin \theta - \sin \theta = 0$

To solve this equation, we can start by factoring out the common term $\sin \theta$ from the left-hand side of the equation.

import math

# Define the equation
def equation(theta):
    return 2 * math.cos(theta) * math.sin(theta) - math.sin(theta)

# Solve for theta
theta = math.asin(1/2)  # Using the arcsine function to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the math.asin function to find the value of $\theta$ that satisfies the equation. The math.asin function returns the arcsine of a given number, which is the angle whose sine is that number. In this case, we use math.asin(1/2) to find the value of $\theta$ that satisfies the equation.

Solution

The value of $\theta$ that satisfies the equation is $\theta = \boxed{26.6}$ degrees.

3) $\sin 2\theta = \frac{1}{2}$

To solve this equation, we can start by using the double-angle identity for sine.

import math

# Define the equation
def equation(theta):
    return math.sin(2 * theta) - 1/2

# Solve for theta
theta = math.asin(1/2) / 2  # Using the arcsine function to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the math.asin function to find the value of $\theta$ that satisfies the equation. The math.asin function returns the arcsine of a given number, which is the angle whose sine is that number. In this case, we use math.asin(1/2) to find the value of $\theta$ that satisfies the equation.

Solution

The value of $\theta$ that satisfies the equation is $\theta = \boxed{15.0}$ degrees.

4) $\cos 2\theta = \frac{1}{2}$

To solve this equation, we can start by using the double-angle identity for cosine.

import math

# Define the equation
def equation(theta):
    return math.cos(2 * theta) - 1/2

# Solve for theta
theta = math.acos(1/2) / 2  # Using the arccosine function to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the math.acos function to find the value of $\theta$ that satisfies the equation. The math.acos function returns the arccosine of a given number, which is the angle whose cosine is that number. In this case, we use math.acos(1/2) to find the value of $\theta$ that satisfies the equation.

Solution

The value of $\theta$ that satisfies the equation is $\theta = \boxed{60.0}$ degrees.

5) $\tan 2\theta = 1$

To solve this equation, we can start by using the double-angle identity for tangent.

import math

# Define the equation
def equation(theta):
    return math.tan(2 * theta) - 1

# Solve for theta
theta = math.atan(1) / 2  # Using the arctangent function to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the math.atan function to find the value of $\theta$ that satisfies the equation. The math.atan function returns the arctangent of a given number, which is the angle whose tangent is that number. In this case, we use math.atan(1) to find the value of $\theta$ that satisfies the equation.

Solution

The value of $\theta$ that satisfies the equation is $\theta = \boxed{22.5}$ degrees.

6) $\sin \theta = \cos \theta$

To solve this equation, we can start by using the identity $\sin \theta = \cos (90^\circ - \theta)$.

import math

# Define the equation
def equation(theta):
    return math.sin(theta) - math.cos(theta)

# Solve for theta
theta = math.pi / 4  # Using the value of pi to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the value of $\pi$ to find the value of $\theta$ that satisfies the equation. The value of $\pi$ is approximately 3.14159.

Solution

The value of $\theta$ that satisfies the equation is $\theta = \boxed{45.0}$ degrees.

7) $\cos \theta = \sin \theta$

To solve this equation, we can start by using the identity $\cos \theta = \sin (90^\circ - \theta)$.

import math

# Define the equation
def equation(theta):
    return math.cos(theta) - math.sin(theta)

# Solve for theta
theta = math.pi / 4  # Using the value of pi to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the value of $\pi$ to find the value of $\theta$ that satisfies the equation. The value of $\pi$ is approximately 3.14159.

Solution

The value of $\theta$ that satisfies the equation is $\theta = \boxed{45.0}$ degrees.

8) $\tan \theta = 1$

To solve this equation, we can start by using the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

import math

# Define the equation
def equation(theta):
    return math.tan(theta) - 1

# Solve for theta
theta = math.pi / 4  # Using the value of pi to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the value of $\pi$ to find the value of $\theta$ that satisfies the equation. The value of $\pi$ is approximately 3.14159.

Solution

The value of $\theta$ that satisfies the equation is $\theta = \boxed{45.0}$ degrees.

9) $\sin \theta = \frac{1}{2}$

To solve this equation, we can start by using the identity $\sin \theta = \frac{1}{2}$.

import math

# Define the equation
def equation(theta):
    return math.sin(theta) - 1/2

# Solve for theta
theta = math.asin(1/2)  # Using the arcsine function to find the value of theta

# Print the value of theta
print("The value of theta is:", round(theta, 1))

In this code, we use the math.asin function to find the value
Grade 11 Trigonometric Equations Revision Exercise 5: Solving Trigonometric Equations

Q&A: Solving Trigonometric Equations

In this article, we will provide answers to some of the most frequently asked questions about solving trigonometric equations.

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions such as sine, cosine, and tangent. These equations are used to solve problems in various fields like physics, engineering, and navigation.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to use various techniques such as factoring, using trigonometric identities, and applying algebraic manipulations. You can also use a calculator or a computer program to solve trigonometric equations.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• $\sec \theta = \frac{1}{\cos \theta}$
• $\csc \theta = \frac{1}{\sin \theta}$

Q: How do I use trigonometric identities to solve an equation?

A: To use trigonometric identities to solve an equation, you need to identify the trigonometric function that is involved in the equation and then use the corresponding identity to simplify the equation.

Q: What is the difference between a trigonometric equation and a trigonometric function?

A: A trigonometric function is a mathematical function that involves trigonometric ratios such as sine, cosine, and tangent. A trigonometric equation, on the other hand, is an equation that involves trigonometric functions.

Q: How do I graph a trigonometric function?

A: To graph a trigonometric function, you need to use a graphing calculator or a computer program. You can also use a table of values to graph a trigonometric function.

Q: What are some common applications of trigonometric equations?

A: Some common applications of trigonometric equations include:

• Physics: Trigonometric equations are used to describe the motion of objects in terms of their position, velocity, and acceleration.
• Engineering: Trigonometric equations are used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Navigation: Trigonometric equations are used to determine the position and velocity of objects in terms of their latitude and longitude.

Q: How do I solve a trigonometric equation with multiple variables?

A: To solve a trigonometric equation with multiple variables, you need to use various techniques such as substitution, elimination, and graphing. You can also use a calculator or a computer program to solve trigonometric equations with multiple variables.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Not using the correct trigonometric identity
• Not simplifying the equation correctly
• Not checking the solution for extraneous solutions
• Not using a calculator or computer program to check the solution

Q: How do I check my solution for extraneous solutions?

A: To check your solution for extraneous solutions, you need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution.

Q: What are some common resources for learning trigonometric equations?

A: Some common resources for learning trigonometric equations include:

• Textbooks: There are many textbooks available that cover trigonometric equations in detail.
• Online resources: There are many online resources available that provide tutorials, examples, and practice problems for trigonometric equations.
• Calculators and computer programs: Calculators and computer programs can be used to solve trigonometric equations and check solutions.

Conclusion

In this article, we have provided answers to some of the most frequently asked questions about solving trigonometric equations. We have also provided some common resources for learning trigonometric equations. With practice and patience, you can become proficient in solving trigonometric equations and apply them to real-world problems.