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7. Example Problems
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Today, we will explore how to use trigonometric ratios to find unknown sides in right-angled triangles. Let’s start with the sine ratio. Can anyone tell me what sine represents?
Isn't it the opposite side over the hypotenuse?
Exactly! Sine of an angle θ is defined as: sin(θ) = Opposite / Hypotenuse. For example, if we have a right triangle with a hypotenuse of 10 cm and an angle of 30°, how do we find the opposite side?
We could use the sine formula. So, Opposite = 10 × sin(30°)?
Correct! And since sin(30°) is 0.5, we find that the opposite side is 5 cm. Great job!
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Now, let’s move on to cosine. Cosine is defined as the adjacent side over the hypotenuse. For instance, if we have a right triangle where the adjacent side is 7 cm and angle θ = 45°, how do we find the hypotenuse?
We can rearrange the cosine formula, right? So Hypotenuse = Adjacent / cos(45°)?
Correct! And since cos(45°) = √2/2, plugging in gives you Hypotenuse = 7 / (√2/2). What does that simplify to?
It simplifies to 7√2, which is about 9.9 cm.
Well done! That’s how we use the cosine ratio to find unknown lengths.
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Create a free accountLet's explore how these trigonometric ratios can apply to real-world problems. For example, if you need to find the height of a tree using its shadow and a measurement of its angle from the ground, what ratios might you use?
We could use the tangent ratio since it is about the opposite side and adjacent side.
Exactly! Using tan(θ) = Opposite / Adjacent allows you to model many real-life situations. Can we think of another situation where we might use sine or cosine?
How about when we're trying to find the length of a ramp for a wheelchair?
Perfect! That’s an excellent application of trig ratios in engineering and accessibility design.
Overview
Short Summary
This section presents practical examples using trigonometric ratios to solve problems involving right-angled triangles.
Medium Summary
The examples provided in this section illustrate the application of trigonometric ratios to calculate unknown sides of right-angled triangles. By working through specific problems, students will gain a deeper understanding of how to apply sine, cosine, and tangent in real-world scenarios.
Detailed Summary
Example Problems
In this section, we delve into practical applications of trigonometric ratios to solve problems involving right-angled triangles. Trigonometric functions such as sine, cosine, and tangent are crucial for linking angles with side lengths in triangles. Here we examine two example problems to illustrate the use of these functions:
Example 1:
A right triangle has an angle 𝜃 = 30° and hypotenuse 10 cm. We want to find the length of the side opposite to angle 𝜃.
- Solution:
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Apply the sine function:
Opposite = Hypotenuse × sin(30°)
Opposite = 10 × 0.5 = 5 cm
Example 2:
In a right-angled triangle, the side adjacent to 𝜃 = 45° is 7 cm. We need to find the hypotenuse.
- Solution:
-
Use the cosine function:
Hypotenuse = Adjacent / cos(45°) = 7 / (√2/2) = 7√2 ≈ 9.9 cm
These examples highlight how trigonometric ratios can be employed to calculate unknown side lengths in right-angled triangles and enhance understanding of the relationships between angles and sides.
Audio Book
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Create a free accountExample 1: A right triangle has an angle 𝜃 = 30∘ and hypotenuse 10 cm. Find the length of the side opposite 𝜃. Solution: 1 Opposite = Hypotenuse×sin30∘ = 10× = 5 cm 2
Detailed Explanation
In this example, we're dealing with a right triangle where one angle (30 degrees) and the length of the hypotenuse (10 cm) are given. We want to find the length of the side opposite the 30-degree angle. To do this, we use the sine function from trigonometry, which relates the opposite side to the hypotenuse through the formula:
Opposite = Hypotenuse × sin(𝜃). By substituting the known values, we get:
Opposite = 10 × sin(30 degrees). The sine of 30 degrees is 0.5, so:
Opposite = 10 × 0.5 = 5 cm. Thus, the length of the side opposite the 30-degree angle is 5 cm.
Examples & Analogies
Imagine a small ramp with an angle of 30 degrees leading up to a slide. If the height of the ramp (the side opposite the angle) needs to be determined and you know the length of the ramp itself (the hypotenuse) is 10 cm, using trigonometry helps you find out that the height, or the side you're interested in, is actually 5 cm high.
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Create a free accountExample 2: In a right-angled triangle, the side adjacent to 𝜃 = 45∘ is 7 cm. Find the hypotenuse. Solution: Adjacent 7 2 Hypotenuse = = = 7× = 7√2 ≈ 9.9 cm cos45∘ √2 √2 2
Detailed Explanation
This example involves a right-angled triangle where we know the length of the adjacent side (7 cm) and the angle (45 degrees). To find the hypotenuse, we use the cosine function, as it relates the adjacent side to the hypotenuse:
Hypotenuse = Adjacent / cos(𝜃). Substituting the known values gives us:
Hypotenuse = 7 / cos(45 degrees). The cosine of 45 degrees is √2 / 2, so substituting that in gives us:
**Hypotenuse = 7 / (√2 / 2) = 7 × (2 / √2) = 7√2 **, which is approximately 9.9 cm.
Examples & Analogies
Think of a ladder leaning against a wall, where the base of the ladder (the adjacent side) is 7 cm away from the wall, and the angle it forms with the ground is 45 degrees. To figure out how tall the ladder reaches (the hypotenuse), we can use the cosine function – just like finding out how far along the ground the ladder is based on the height it reaches at the wall.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Trigonometric Ratios: Used to determine relationships between angles and sides in right-angled triangles.
Sine, Cosine, and Tangent: Fundamental functions that relate angles to side lengths.
Applying Ratios: Understanding how to practically apply these ratios to find unknown side lengths and angles.
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
When calculating the opposite side length using sin(30°), if the hypotenuse is 10 cm, the opposite side is 5 cm.
In a triangle where the adjacent side measures 7 cm and θ = 45°, the hypotenuse is calculated using cosine to be approximately 9.9 cm.
Memory Aids
Interactive tools to help you remember key concepts
Stories
Memory Tools
Flash Cards
Glossary
Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Opposite Side
The side opposite to the angle of interest (θ) in a right-angled triangle.
Adjacent Side
The side next to the angle of interest (θ) and not the hypotenuse.
Sine Ratio
The ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
Cosine Ratio
The ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle.
Tangent Ratio
The ratio of the length of the opposite side to the adjacent side in a right-angled triangle.