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8.3. Worked example
Interactive Audio Lesson
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Today, we will examine right-angled triangles and how we can apply trigonometric ratios in them. Who can tell me what the Pythagorean theorem states?
Isn't it a² + b² = c², where c is the hypotenuse?
Exactly! This relationship is vital in our worked example today. So, in triangle ABC, if AB is the hypotenuse, and we know that BC is one side, how do we find AC?
We can use the Pythagorean theorem!
That's right! Let's calculate AC together.
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Using our values, AB equals 13 and BC equals 12. Let's find AC: AC = √(AB² - BC²). Any volunteers to do this calculation?
I can do it! So, AC = √(13² - 12²) = √(169 - 144), which equals √25 = 5.
Perfect! Now we have AC. Let’s move on to finding sin(A). Who knows the formula for sine?
It’s sine = opposite over hypotenuse, right?
Exactly! Now, what is opposite to angle A?
That's AC, which we found equals 5!
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Create a free accountGood job! Now to calculate sin(A): sin(A) = AC/AB. What is that equal to?
It’s 5/13, which is approximately 0.3846!
Excellent! This shows us how we can use the lengths in right triangles not just to find missing sides, but also to find angles. Recap for us the steps we took in this example.
First, we used the Pythagorean theorem to find AC, and then we used the sine ratio to find sin(A).
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Create a free accountNow that we have completed this worked example, think about how you could apply a similar approach to other right-angle triangles. Can anyone provide a scenario where you might need to use the sine function?
Maybe in architecture, if I need to know the height of a building?
Exactly! Knowing how to calculate these values can be crucial in many real-world situations. Let’s summarize: What key concepts did we learn today?
We learned how to find a side length using the Pythagorean theorem and how to calculate sine using the ratios of sides.
Overview
Short Summary
This worked example illustrates how to apply trigonometric ratios in a right triangle to find unknown side lengths and angles.
Medium Summary
In this worked example, we focus on triangle ABC, which is right-angled at vertex C. We use the Pythagorean theorem and sine function to find the length of side AC and the sine of angle A. This example highlights the connection between side lengths and angles in trigonometric calculations.
Detailed Summary
In the worked example from Section 8.3, we analyze triangle ABC, which is right-angled at C. Given that side AB equals 13 units and side BC equals 12 units, we need to find the length of side AC and the sine of angle A. To find AC, we apply the Pythagorean theorem: AC = √(AB² - BC²). This results in AC = √(13² - 12²) = √(169 - 144) = √25 = 5 units. To find sin(A), we use the definition of the sine function: sin(A) = opposite/hypotenuse = AC/AB = 5/13, which is approximately equal to 0.3846. This example emphasizes the critical role of trigonometric ratios in solving problems involving right triangles, serving as foundational knowledge for more advanced topics.
Audio Book
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Create a free accountIn ΔABC right-angled at C, with side AB = 13, BC = 12, find AC and sin(A).
AC = √(13² − 12²) = 5.
Detailed Explanation
In triangle ABC, we know that this triangle is right-angled at point C. According to the Pythagorean theorem, the squares of the lengths of the two shorter sides (BC and AC) add up to the square of the longest side (AB, which is the hypotenuse). Since AB = 13 and BC = 12, we can set up the equation:
- Calculate the square of AB: 13² = 169.
- Calculate the square of BC: 12² = 144.
- Using the Pythagorean theorem, we find that AC² = AB² - BC², which means AC² = 169 - 144.
- This simplifies to AC² = 25.
- Taking the square root gives us AC = 5.
Examples & Analogies
Imagine you have a ladder that is 13 feet long (AB), leaning against a wall (which creates the right triangle). If the bottom of the ladder is 12 feet from the wall (BC), you can use the Pythagorean theorem to find out how high the ladder reaches on the wall (AC). By calculating, you find it's 5 feet high!
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Create a free accountsin(A) = 5 / 13 ≈ 0.3846.
Detailed Explanation
To find sin(A), we utilize the definition of sine in a right triangle. The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle (here side AC) to the length of the hypotenuse (side AB). We already calculated AC = 5 and know AB = 13. Therefore:
- sin(A) = opposite side (AC) / hypotenuse (AB).
- Substituting the lengths gives us sin(A) = 5 / 13.
- Calculating this yields approximately 0.3846.
Examples & Analogies
Think of angle A as a ramp leading to a loading dock. If the ramp rises 5 feet (the height), and the length of the ramp itself is 13 feet, sin(A) represents how steep the ramp is. A steeper ramp would have a higher sine value, while a flatter ramp would have a lower value.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Pythagorean Theorem: Relates the sides of a right triangle through a² + b² = c².
Sine Function: The ratio of the length of the opposite side to the hypotenuse.
Right Triangle Properties: Includes definitions and applications of the right triangle.
Examples
Memory Aids
Interactive tools to help you remember key concepts
Stories
Memory Tools
Flash Cards
Glossary
Pythagorean Theorem
A mathematical principle stating that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Sine Function (sin)
A trigonometric function that relates an angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse.
Right Triangle
A triangle that has one angle equal to 90 degrees.