Hey guys! Let's dive into the solutions for Exercise 26 from Unit 2 of your Grade 11 Maths textbook. I know maths can sometimes feel like climbing a mountain, but don't worry, we'll tackle it together step by step. This unit typically covers topics like trigonometry, functions, and coordinate geometry, so expect some problems related to these areas. Remember, the key to mastering maths is practice, practice, and more practice! So grab your notebooks, pens, and let's get started!

Understanding the Core Concepts

Before we jump right into the solutions, let's quickly recap the main concepts covered in Unit 2. This will help us approach the problems in Exercise 26 with a solid foundation. We're generally talking about trigonometric identities, understanding angles and their measurements in radians and degrees, and the relationships between trigonometric functions. We also need to be comfortable with graphing trigonometric functions and solving trigonometric equations. Let's break it down further:

• Trigonometric Identities: These are equations that are true for all values of the variables involved. Common identities include the Pythagorean identities (sin²θ + cos²θ = 1), reciprocal identities (csc θ = 1/sin θ), and quotient identities (tan θ = sin θ/cos θ). Understanding and memorizing these identities is crucial for simplifying expressions and solving equations.
• Angles and Their Measurements: We need to be fluent in converting between degrees and radians. Remember that π radians is equal to 180 degrees. Also, understanding coterminal angles (angles that share the same terminal side) is important.
• Trigonometric Functions: This involves understanding the definitions of sine, cosine, tangent, cosecant, secant, and cotangent in terms of the sides of a right-angled triangle. We also need to know the unit circle definitions and how the signs of these functions vary in different quadrants.
• Graphs of Trigonometric Functions: Knowing the basic shapes of the graphs of sine, cosine, and tangent is essential. We should also be able to identify the amplitude, period, phase shift, and vertical shift of these graphs.
• Solving Trigonometric Equations: This involves using trigonometric identities and algebraic techniques to find the values of the variable that satisfy the equation. Remember to consider the general solution, which includes all possible solutions within a given interval.

Having a strong grasp of these core concepts will make solving the problems in Exercise 26 much easier. If you're feeling a bit rusty, I recommend reviewing your textbook or online resources before proceeding.

Tackling Exercise 26: A Step-by-Step Approach

Okay, guys, now that we've refreshed our understanding of the core concepts, let's get down to business and tackle Exercise 26. Since I don't have the exact questions from your textbook, I'll create some typical problems that you might encounter and provide detailed solutions. Remember, the goal is not just to get the right answer but to understand the process involved. So, let's dive in!

Problem 1:

Simplify the expression: (sin θ + cos θ)² + (sin θ - cos θ)²

Solution:

1. Expand the squares:

   (sin² θ + 2sin θ cos θ + cos² θ) + (sin² θ - 2sin θ cos θ + cos² θ)

2. Combine like terms:

   2sin² θ + 2cos² θ

3. Factor out the 2:

   2(sin² θ + cos² θ)

4. Apply the Pythagorean identity (sin² θ + cos² θ = 1):

   2(1) = 2

Therefore, the simplified expression is 2.

Problem 2:

Solve the equation: 2cos θ - 1 = 0, for 0 ≤ θ ≤ 2π

Solution:

1. Isolate cos θ:

   2cos θ = 1 cos θ = 1/2

2. Find the angles whose cosine is 1/2 within the given interval:

   θ = π/3 and θ = 5π/3

Therefore, the solutions are θ = π/3 and θ = 5π/3.

Problem 3:

Sketch the graph of y = 2sin(x - π/2)

Solution:

1. Identify the key parameters:

   • Amplitude: 2
   • Period: 2π
   • Phase shift: π/2 to the right

2. Start with the basic sine graph (y = sin x).

3. Apply the amplitude: Stretch the graph vertically by a factor of 2.

4. Apply the phase shift: Shift the graph π/2 units to the right.

The resulting graph will be a sine wave with an amplitude of 2, a period of 2π, and shifted π/2 units to the right.

Problem 4:

Prove the identity: (1 + tan² θ)cos² θ = 1

Solution:

1. Start with the left-hand side of the equation:

   (1 + tan² θ)cos² θ

2. Use the identity tan² θ = sin² θ/cos² θ:

   (1 + sin² θ/cos² θ)cos² θ

3. Distribute the cos² θ:

   cos² θ + sin² θ

4. Apply the Pythagorean identity (sin² θ + cos² θ = 1):

   1

Therefore, the left-hand side is equal to the right-hand side, proving the identity.

Tips and Tricks for Success

Alright, guys, before you head off to conquer Exercise 26, here are a few extra tips and tricks to keep in mind:

• Master the Identities: Make sure you have a solid understanding of all the trigonometric identities. Create flashcards or a cheat sheet to help you memorize them.
• Practice Regularly: The more you practice, the more comfortable you'll become with solving trigonometric problems. Set aside some time each day to work on maths problems.
• Draw Diagrams: When dealing with angles and trigonometric functions, drawing diagrams can be incredibly helpful. This will help you visualize the relationships between the different quantities.
• Check Your Answers: Always check your answers to make sure they make sense. Substitute your solutions back into the original equation to verify that they are correct.
• Don't Be Afraid to Ask for Help: If you're struggling with a particular problem, don't hesitate to ask your teacher, classmates, or online resources for help. There's no shame in admitting that you need assistance.

Conclusion

So there you have it, guys! A comprehensive guide to tackling Exercise 26 from Unit 2 of your Grade 11 Maths textbook. Remember, maths is all about understanding the underlying concepts and practicing regularly. With a little bit of effort and dedication, you can master trigonometry and ace your exams. Good luck, and happy solving!

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