Interactive Audio Lesson
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Proving Trigonometric Identities
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Today, we will begin with proving a fundamental identity: 1 + tanยฒฮธ = secยฒฮธ. Can anyone remind me what tanยฒฮธ is in terms of sine and cosine?
Isn't tanยฒฮธ equal to sinยฒฮธ/cosยฒฮธ?
Exactly! So how can we rewrite 1 + tanยฒฮธ using that definition?
We can write 1 as cosยฒฮธ/cosยฒฮธ, then it becomes cosยฒฮธ/cosยฒฮธ + sinยฒฮธ/cosยฒฮธ.
Perfect! What do you get when you combine those fractions?
It becomes (cosยฒฮธ + sinยฒฮธ)/cosยฒฮธ, and since cosยฒฮธ + sinยฒฮธ = 1, it simplifies to 1/cosยฒฮธ.
Well done! So we have shown 1 + tanยฒฮธ = secยฒฮธ. Remember, the key is knowing those foundational identities.
Thank you! This really helps clarify how the identities work together.
Finding Trigonometric Function Values
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Next, let's tackle the problem: If sin(x) = 3/5 and x is in the first quadrant, can anyone find cos(x) and tan(x)?
To find cos(x), I can use the Pythagorean identity, right?
Exactly! Can you show us how?
So, using sinยฒx + cosยฒx = 1, I calculate cosยฒx = 1 - (3/5)ยฒ, which is 1 - 9/25, giving me cosยฒx = 16/25?
Yes, and what is cos(x) then, since x is in the first quadrant?
That means cos(x) = โ(16/25) = 4/5.
Great! Now, how do you find tan(x)?
tan(x) = sin(x)/cos(x), so it's (3/5)/(4/5) which simplifies to 3/4!
Excellent work! You've successfully found all the required values.
Sketching Graphs
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Now, let's move to graphing. Sketch the graph of y = sin(x - 90ยฐ). What transformation do we have here?
Thatโs a phase shift to the right by 90ยฐ!
Correct! What would the key point at x = 0 be?
At x = 0, y = sin(-90ยฐ), which equals -1.
And how about at x = 360ยฐ?
At x = 360ยฐ, y = sin(270ยฐ), which is also -1!
Fantastic! Now, plot this key point and identify other key points to complete your graph.
Got it! This is fun!
Solving Equations
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Let's tackle the equation: 2sinยฒx - 1 = 0. Can someone solve this for 0ยฐ โค x โค 360ยฐ?
I can factor it as sinยฒx = 1/2, which means sin(x) = ยฑโ(1/2).
That's right! What angles are solutions for sin(x) = โ(1/2)?
x = 45ยฐ and 135ยฐ.
Good job! And what about sin(x) = -โ(1/2)?
That would give us x = 225ยฐ and 315ยฐ!
Excellent! So the complete solutions are?
x = 45ยฐ, 135ยฐ, 225ยฐ, and 315ยฐ!
Understanding Amplitude and Period
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Lastly, letโs analyze the function y = โ2cos(3x). What can you tell me about its amplitude and period?
The amplitude is the absolute value of -2, which is 2.
Correct! And how is the period calculated?
We calculate the period using the formula 360ยฐ / B, so the period is 360ยฐ / 3, which equals 120ยฐ.
Excellent! Now, how will the graph differ due to the negative amplitude?
It will be reflected over the x-axis, right?
Exactly! You've grasped the transformations nicely.
Introduction & Overview
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Quick Overview
Standard
In this section, students will tackle various practice problems focusing on trigonometric identities, the values of trigonometric functions in different scenarios, and graph transformations. These exercises are designed to enhance problem-solving skills and solidify the concepts learned in the chapter.
Detailed
Practice Problems
This section is dedicated to reinforcing the concepts taught in the chapter on Trigonometric Identities and Graphs. The practice problems cover essential trigonometric identities, relationships between functions, and transformations of trigonometric graphs.
The problems are categorized into various types including proving identities, finding function values based on given conditions, sketching graphs based on transformations, and solving equations. Working through these problems will help students apply their understanding of trigonometric functions and prepare them for more complex mathematical challenges. Furthermore, the variety of problems encourages deeper engagement with the material, promoting critical thinking and analytical skills.
Audio Book
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Prove the Identity
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- Prove: 1 + tanยฒฮธ = secยฒฮธ
Detailed Explanation
To prove this identity, recall that tan(ฮธ) = sin(ฮธ)/cos(ฮธ) and sec(ฮธ) = 1/cos(ฮธ). Therefore, tanยฒ(ฮธ) = sinยฒ(ฮธ)/cosยฒ(ฮธ). By substituting this into the equation 1 + tanยฒฮธ, we get:
1 + tanยฒฮธ = 1 + (sinยฒฮธ/cosยฒฮธ) = (cosยฒฮธ/cosยฒฮธ) + (sinยฒฮธ/cosยฒฮธ) = (sinยฒฮธ + cosยฒฮธ)/cosยฒฮธ. Since we know by the Pythagorean identity that sinยฒฮธ + cosยฒฮธ = 1, we have:
(sinยฒฮธ + cosยฒฮธ)/cosยฒฮธ = 1/cosยฒฮธ = secยฒฮธ, thus proving the identity.
Examples & Analogies
Think of this as verifying a recipe for a cake. You have the final result (the cake) and need to show that all the ingredients used (the values of sin and cos) correctly combine to give the final product (the identity). This step-by-step validation ensures that what you've crafted is indeed correct, just like in mathematics.
Finding Cosine and Tangent
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- If sin(x) = 3/5 and x is in the first quadrant, find cos(x) and tan(x)
Detailed Explanation
To find cos(x) and tan(x) when sin(x) = 3/5, we can use the Pythagorean identity sinยฒฮธ + cosยฒฮธ = 1. Substitute sin(x):
(3/5)ยฒ + cosยฒ(x) = 1
9/25 + cosยฒ(x) = 1
cosยฒ(x) = 1 - 9/25 = 16/25
Then, cos(x) = โ(16/25) = 4/5.
Now, to find tan(x), recall that tan(x) = sin(x)/cos(x):
tan(x) = (3/5) / (4/5) = 3/4.
Examples & Analogies
Imagine you are climbing a ladder. If the height you reach (representing sin) is 3 rungs out of a total of 5, the total distance from the base (representing the hypotenuse) is your ladder. By determining how far horizontally you have stretched (the cosine), you can now figure out your steepness or 'angle' (the tangent) of the ladderโfinding how your vertical height relates to your horizontal distance.
Sketching a Graph
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- Sketch the graph of y = sin(x โ 90ยฐ)
Detailed Explanation
The equation y = sin(x - 90ยฐ) indicates a phase shift of 90ยฐ to the right. Since the sine graph typically starts at the origin (0,0), beginning from x = 90ยฐ shifts the peak of the sine wave to the y-axis point (0,1). This phase shift alters the graph's starting point but maintains the wave's shape and amplitude at a value of 1. Thus, key points on the graph should be drawn considering this shift.
Examples & Analogies
Imagine setting an alarm for 7 AM, but you want to wake up at 8 AM instead. By physically moving the time of your alarm (shifting the graph), the way your morning routine looks (the entire sine wave) doesnโt change in its nature, but it starts at a different pointโjust as sin(x) shifts with phase changes.
Solving an Equation
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- Solve the equation: 2 sinยฒx โ 1 = 0 for 0ยฐ โค x โค 360ยฐ
Detailed Explanation
To solve 2 sinยฒx - 1 = 0, first isolate sinยฒx:
2 sinยฒx = 1
sinยฒx = 1/2.
Taking the square root gives us sin(x) = โ(1/2) or sin(x) = -โ(1/2). Since we need solutions within the first revolution 0ยฐ to 360ยฐ, we find:
sin(x) = โ(1/2) โ x = 45ยฐ, 135ยฐ sin(x) = -โ(1/2) โ x = 225ยฐ, 315ยฐ.
Therefore, the solutions are x = 45ยฐ, 135ยฐ, 225ยฐ, 315ยฐ.
Examples & Analogies
Think of this as trying to find out at what times during a day the sun is at a specific brightness level (the height of the sine function). Youโd look at the times when itโs exactly halfway up before noon or after noon, realizing that there are multiple points in the day when that brightness occurs, similar to finding multiple solutions in your equation.
Amplitude and Period
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- What is the amplitude and period of y = โ2 cos(3x)?
Detailed Explanation
In the function y = -2 cos(3x), the amplitude is the absolute value of the coefficient in front of cos, which is |-2| = 2. This value represents how high or low the graph reaches from the center line.
The period of a cosine function is calculated using the formula: Period = 360ยฐ / B, where B is the coefficient of x. Here, B = 3, so:
Period = 360ยฐ / 3 = 120ยฐ.
Examples & Analogies
Think of a trampoline that can bounce 2 units high and low around the resting point (the center line of the graph). The way you bounce affects how often you complete a cycle (the period)โif you bounce faster (with a higher frequency), youโre completing more bounces in the same timeframe, just like a function completing its wave pattern more quickly.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Trigonometric Identities: Equations like Pythagorean identities are essential for simplification and proofs.
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Finding Function Values: Skills in determining values of sin, cos, and tan from given relationships.
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Graph Transformations: Understanding how phase shifts and vertical shifts impact the shape and position of graphs.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of proving the identity 1 + tanยฒฮธ = secยฒฮธ using known values.
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Finding the values of cos(x) and tan(x) when sin(x) = 3/5 in the first quadrant.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
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Sine and Cosine, they dance in pairs, their squares equal one; it's a truth so rare.
๐ Fascinating Stories
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Imagine a wave at a beach. The height it rises is called amplitude, while how far it travels before returning is its period. They play together in the rhythm of the ocean.
๐ง Other Memory Gems
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For sine and cosine, remember itโs SOH-CAH-TOA! Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.
๐ฏ Super Acronyms
P.A.S.S. for Graph Transformation
- P: - Phase Shift
- A: - Amplitude
- S: - Shape changes
- S: - Shift vertically.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Trigonometric Identity
Definition:
An equation involving trigonometric functions that holds true for all values of the variable.
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Term: Amplitude
Definition:
The maximum height of a wave from its center line.
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Term: Period
Definition:
The distance between repeating points in a wave; in trigonometric functions, it's measured in degrees.
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Term: Phase Shift
Definition:
A horizontal shift in the graph of a function.
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Term: Vertical Shift
Definition:
A vertical movement of the graph, up or down.