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5.2 - Trigonometric Ratios

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Introduction to Trigonometric Ratios

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Teacher
Teacher

Today, we're going to learn about trigonometric ratios! They are key to understanding the relationship between angles and sides in right-angled triangles. Can anyone tell me what we know about right triangles?

Student 1
Student 1

They have one angle that's 90 degrees, and the longest side is called the hypotenuse!

Teacher
Teacher

Exactly! And there are also two other sides, the opposite side and the adjacent side. Can anyone recall how we define sine, cosine, and tangent?

Student 2
Student 2

Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent!

Teacher
Teacher

Perfect! To remember this, you can use the acronym SOH-CAH-TOA. Who remembers what that stands for?

Student 3
Student 3

SOH means sine equals opposite over hypotenuse, CAH means cosine equals adjacent over hypotenuse, and TOA means tangent equals opposite over adjacent!

Teacher
Teacher

Great! Remembering these ratios will help you solve many problems. Let's summarize these concepts before moving on.

Reciprocal Trigonometric Functions

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Teacher
Teacher

In addition to the primary ratios, we also have reciprocal functions. Can someone tell me what they are?

Student 4
Student 4

Cosecant, secant, and cotangent!

Teacher
Teacher

Correct! Can anyone tell me how to define those?

Student 1
Student 1

Cosecant is 1/sine, secant is 1/cosine, and cotangent is 1/tangent!

Teacher
Teacher

Exactly! Remember, these can be helpful in various calculations. Let’s look at how to apply these ratios practically in finding sine and cosine values for specific angles.

Example Application of Trigonometric Ratios

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Teacher
Teacher

Let's work through an example together. If we have a right triangle where the opposite side to angle 30 degrees is 5 cm, and the hypotenuse is 10 cm, how do we find sin and cos of 30 degrees?

Student 2
Student 2

For sine, it's 5 over 10, which simplifies to 1/2.

Teacher
Teacher

Correct! And what about cosine?

Student 3
Student 3

We need to find the length of the adjacent side first using the Pythagorean theorem, right?

Teacher
Teacher

Yes, well done! The adjacent side is √(10² - 5²), which is √75 or 5√3 cm. So what is cos 30 degrees?

Student 4
Student 4

Cosine will be 5√3 over 10, which simplifies to √3/2.

Teacher
Teacher

Yes, excellent! This shows how trigonometric ratios are used in real calculations. Always remember to write down your ratios clearly and use the triangle sides accurately.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section introduces trigonometric ratios that relate the angles and sides of right triangles, focusing on sine, cosine, and tangent.

Standard

In this section, we explore trigonometric ratios for acute angles in right-angled triangles, defining sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent. Additionally, an example illustrates how to apply these concepts to calculate specific trigonometric values.

Detailed

Detailed Summary

In this section, we delve into trigonometric ratios, which are vital relationships that assist in defining the angles and sides of right-angled triangles. For an acute angle θ, the following ratios are defined:

  • Sine (sin θ) as the ratio of the length of the opposite side to the length of the hypotenuse:

$$
ext{sin }θ = \frac{\text{Opposite}}{\text{Hypotenuse}}
$$
- Cosine (cos θ) as the ratio of the length of the adjacent side to the length of the hypotenuse:

$$
ext{cos }θ = \frac{\text{Adjacent}}{\text{Hypotenuse}}
$$
- Tangent (tan θ) as the ratio of the length of the opposite side to the length of the adjacent side:

$$
ext{tan }θ = \frac{\text{Opposite}}{\text{Adjacent}}
$$

Additionally, the reciprocal functions are defined as follows:
- Cosecant (csc θ) = 1/sin θ
- Secant (sec θ) = 1/cos θ
- Cotangent (cot θ) = 1/tan θ

An example illustrated in this section walks through a practical application of these ratios, where given specific lengths in a right triangle, students are tasked with finding the sine and cosine values for 30 degrees. This practical approach helps in grounding the definitions and offers real-world relevance to trigonometric ratios.

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Audio Book

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Definition of Trigonometric Ratios

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For an acute angle θ, in a right-angled triangle:
- sin θ = Opposite / Hypotenuse
- cos θ = Adjacent / Hypotenuse
- tan θ = Opposite / Adjacent

Detailed Explanation

Trigonometric ratios relate the angles and sides of a right triangle. For a given acute angle θ, these ratios are used to express the relationships of the sides opposite and adjacent to the angle compared to the hypotenuse (the longest side of the triangle). The sine (sin) ratio gives the length of the opposite side divided by the hypotenuse, the cosine (cos) ratio gives the length of the adjacent side divided by the hypotenuse, and the tangent (tan) ratio gives the length of the opposite side divided by the adjacent side.

Examples & Analogies

Imagine you're using a ladder against a wall. The angle the ladder makes with the ground is like our angle θ. The height of the wall represents the 'opposite' side, the distance from the wall to the base of the ladder represents the 'adjacent' side, and the ladder itself is the 'hypotenuse.' By knowing one of these sides and the angle, you can use trigonometric ratios to figure out the others!

Reciprocal Trigonometric Ratios

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Also,
- cot θ = 1 / tan θ
- sec θ = 1 / cos θ
- csc θ = 1 / sin θ

Detailed Explanation

In addition to the primary trigonometric ratios, there are reciprocal identities that are important to understand. The cotangent (cot) ratio is the reciprocal of the tangent, meaning if you take the tangent of the angle and flip it over, you'll get the cotangent. Similarly, the secant (sec) is the reciprocal of the cosine, and the cosecant (csc) is the reciprocal of the sine. These relationships allow for flexibility in solving problems as they offer alternatives based on different side measurements.

Examples & Analogies

Think of reciprocal relationships in reality, like when you are measuring something from different perspectives. If you only know how tall something is (like a tree) and you are standing a certain distance away from it, you can use these ratios to find new insights about its height or your distance from it, just like flipping numbers in division gives us different values!

Practical Example of Trigonometric Ratios

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In a right triangle, if the length of the side opposite to angle 30° is 5 cm and the hypotenuse is 10 cm, find sin 30° and cos 30°.
- Solution:
- sin 30° = Opposite / Hypotenuse = 5 / 10 = 1 / 2
- Using Pythagoras theorem, the adjacent side = √(10² - 5²) = √(100 - 25) = √(75) = 5√3 cm.
- cos 30° = Adjacent / Hypotenuse = (5√3) / 10 = √3 / 2.

Detailed Explanation

To find the trigonometric ratios for angle 30° in this example, we first use the sine ratio. We know the length of the opposite side (5 cm) and the hypotenuse (10 cm), so we calculate sin 30° by dividing these two sides, giving us 1/2. Next, we need to find the length of the adjacent side. We apply the Pythagorean theorem, which tells us that the square of the hypotenuse (10²) is equal to the sum of the squares of the other two sides (opposite² + adjacent²). We calculate the adjacent side as √(100 - 25) = 5√3. Finally, we find cos 30° using the ratio of the adjacent side (5√3 cm) to the hypotenuse (10 cm), which simplifies to √3/2.

Examples & Analogies

If you've ever tried to measure how tall a building is by using a ladder, the measurements you use to find the angle of the ladder against the ground can also relate to angles in a right triangle. This example shows that even when measuring the building's height directly can be tricky, using trigonometric ratios can help find out its height based on your position and the angle you make with the ladder.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Trigonometric Ratios: Sine, Cosine, and Tangent which relate angle to sides in right triangles.

  • Reciprocal Functions: Cosecant, Secant, and Cotangent derived from primary trigonometric ratios.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example: For a right triangle with an opposite side of 5 cm and a hypotenuse of 10 cm, sin 30° = 5/10 = 1/2 and cos 30° can be found using Pythagorean theorem, resulting in √3/2.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Sin is opposite, cos is near, tan's the ratio we hold dear.

📖 Fascinating Stories

  • Once upon a time, in a triangle, sine loved to go high, opposite its friend, while cosine was always side by side. And tangent, the tricky one, would jump right over, always comparing the opposite to adjacent, just like a roller coaster!

🧠 Other Memory Gems

  • SOH-CAH-TOA: SOH (Sine = Opposite/Hypotenuse), CAH (Cosine = Adjacent/Hypotenuse), TOA (Tangent = Opposite/Adjacent).

🎯 Super Acronyms

ROCK

  • Reciprocal Of Cosine = Ksec
  • which reminds us secant is the inverse of the cosine function.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Sine

    Definition:

    The ratio of the length of the opposite side to the hypotenuse in a right triangle.

  • Term: Cosine

    Definition:

    The ratio of the length of the adjacent side to the hypotenuse in a right triangle.

  • Term: Tangent

    Definition:

    The ratio of the length of the opposite side to the adjacent side in a right triangle.

  • Term: Cosecant

    Definition:

    The reciprocal of sine; equal to 1/sin θ.

  • Term: Secant

    Definition:

    The reciprocal of cosine; equal to 1/cos θ.

  • Term: Cotangent

    Definition:

    The reciprocal of tangent; equal to 1/tan θ.