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8.3 - Trigonometric Ratios of Some Specific Angles

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Understanding Trigonometric Ratios of 45°

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Teacher
Teacher

Today we will explore the trigonometric ratios for 45°. Can anyone tell me about the properties of a triangle when one of the angles is 45°?

Student 1
Student 1

In a right triangle, if one angle is 45°, the other angle must also be 45°.

Teacher
Teacher

Exactly! This creates an isosceles right triangle where the two sides are equal. Now, if we denote those sides as 'a', how would we calculate the hypotenuse?

Student 2
Student 2

We use the Pythagorean theorem: AC² = AB² + BC², which means AC = a√2.

Teacher
Teacher

Correct! Now using this, can anyone derive sin 45°, cos 45°, and tan 45°?

Student 3
Student 3

Yes! Sin 45° = a / (a√2) = 1/√2, which is √2/2. Cos 45° will be the same, and tan 45° = 1.

Teacher
Teacher

Perfect! Remember the acronym 'SCC' for 'Sine, Cosine, Complementary' to help you recall that sin and cos are equal at 45°.

Teacher
Teacher

Let's summarize: For 45°, we have sin 45° = √2/2, cos 45° = √2/2, and tan 45° = 1.

Exploring 30° and 60° Trigonometric Ratios

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Teacher
Teacher

Now, let’s discuss 30° and 60°. Who can tell me the relationship between these angles?

Student 2
Student 2

They are complementary angles, meaning they add up to 90°.

Teacher
Teacher

Exactly! We can derive their ratios using an equilateral triangle. If each side is 2a, what can we conclude?

Student 4
Student 4

We draw a height from one vertex to the opposite side, creating two 30-60-90 triangles.

Teacher
Teacher

Well done! In this scenario, how do we calculate sin 30°?

Student 1
Student 1

Sin 30° = opposite / hypotenuse, which is a / 2a = 1/2.

Teacher
Teacher

And for sin 60°?

Student 3
Student 3

Sin 60° = √3/2 because the opposite side is a√3 and the hypotenuse is 2a.

Teacher
Teacher

Great! The key angles to remember: sin 30° = 1/2, sin 60° = √3/2. Now let’s summarize.

Understanding Trigonometric Values for 0° and 90°

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Teacher
Teacher

Let’s finalize by discussing sin and cos for 0° and 90°. What can we observe as we approach these angles?

Student 4
Student 4

As the angle approaches 0°, the opposite side length approaches 0, which makes sin 0° = 0.

Teacher
Teacher

Exactly! And what about cos 0°?

Student 1
Student 1

Cos 0° = 1 since the hypotenuse remains unchanged.

Teacher
Teacher

Right! Now, as we move towards 90°, what happens to sin and cos?

Student 2
Student 2

Sin 90° becomes 1, and cos 90° becomes 0 as the triangle squashes.

Teacher
Teacher

Well summarized! For 0°: sin 0° = 0, cos 0° = 1; and for 90°: sin 90° = 1, cos 90° = 0. Let’s recap that!

Introduction & Overview

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

Quick Overview

This section explores the values of trigonometric ratios for the angles of 0°, 30°, 45°, 60°, and 90° using geometric constructions.

Standard

Students will learn to calculate trigonometric ratios for specific angles like 0°, 30°, 45°, 60°, and 90°. This section derives these values through geometric constructions and the properties of right triangles, emphasizing the significance in trigonometry.

Detailed

Detailed Summary

In this section, we delve into the trigonometric ratios of key angles of 0°, 30°, 45°, 60°, and 90° through the application of right triangle properties and geometric constructions. We begin with calculating ratios for 45°, highlighting that in a right triangle where both angles are 45°, the legs are equal in length. By utilizing the Pythagorean theorem, we formulate the ratios, demonstrating that:
- sin 45° = cos 45° = √2/2
- tan 45° = 1

Next, we explore the ratios for angles 30° and 60° using an equilateral triangle. By drawing a perpendicular from one vertex, we create two right triangles, showing that:
- sin 30° = 1/2 and sin 60° = √3/2
- cos 30° = √3/2 and cos 60° = 1/2
- tan 30° = 1/√3 and tan 60° = √3

Finally, we address the limits of trigonometric functions as angles approach 0° and 90°, defining values for:
- sin 0° = 0, cos 0° = 1
- sin 90° = 1, cos 90° = 0

The section concludes by summarizing the values of all calculated trigonometric ratios in a convenient table for reference.

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Audio Book

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Trigonometric Ratios of 45°

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In Δ ABC, right-angled at B, if one angle is 45°, then the other angle is also 45°, i.e., ∠ A = ∠ C = 45° (see Fig 8.14).

So, BC = AB (Why?)

Now, Suppose BC = AB = a.

Then by Pythagoras Theorem, AC² = AB² + BC² = a² + a² = 2a², and, therefore, AC = a√2.

Using the definitions of the trigonometric ratios, we have:
- sin 45° = BC/AC = a / (a√2) = 1/√2
- cos 45° = AB/AC = a / (a√2) = 1/√2
- tan 45° = BC/AB = a/a = 1
- cosec 45° = 1/sin 45° = √2,
- sec 45° = 1/cos 45° = √2,
- cot 45° = 1/tan 45° = 1.

Detailed Explanation

In a right triangle where one angle is 45°, both other angles must also be 45° making this a special isosceles triangle. The lengths of the two legs (BC and AB) are equal. By using the Pythagorean theorem, we find the length of the hypotenuse (AC) as 'a√2', given that both legs are of equal length 'a'. The trigonometric ratios can then be calculated, revealing that sine and cosine of 45° are both equal to '1/√2', which simplifies to approximately 0.707.

Examples & Analogies

Imagine you're climbing a staircase where each step makes a 45° angle with the ground. Since every time you go up one step (the height), you also move out one step (the horizontal distance), both distances are equal. If you used a ruler to measure these steps, both would be the same, reinforcing that the sine and cosine of those angles are equal.

Trigonometric Ratios of 30° and 60°

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Let us now calculate the trigonometric ratios of 30° and 60°. Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore, ∠ A = ∠ B = ∠ C = 60°.

Draw the perpendicular AD from A to the side BC (see Fig. 8.15).

Now, Δ ABD ≅ Δ ACD (Why?) Thus, BD = DC and ∠ BAD = ∠ CAD (CPCT).

Now observe that:
Δ ABD is a right triangle, right-angled at D with ∠ BAD = 30° and ∠ ABD = 60°.
As you know, for finding the trigonometric ratios, we need to know the lengths of the sides of the triangle. So, let us suppose that AB = 2a.

Then, BD = BC= a/2 and AD² = AB² – BD² = (2a)² – (a)² = 3a², Therefore, AD = a√3. Now, we have:
- sin 30° = BD/AB = a / (2a) = 1/2,
- cos 30° = AD/AB = (a√3)/(2a) = √3/2,
- tan 30° = BD/AD = a/(a√3) = 1/√3.

Detailed Explanation

In the triangle, we derive the lengths needed to calculate the trigonometric ratios for 30° and 60°. Since we form a right triangle from half of the equilateral triangle, we can use properties of the angles and the lengths of the sides to derive sine and cosine values specifically tied to these angles. The side relationships reveal that 30° has a sine of 1/2 and a cosine of √3/2, while the tangent is 1/√3, making them interconnected using the sides drawn from a common height.

Examples & Analogies

Think of a tall tree. If you measure the height of the tree and find a point where the angle from your eyes to the top of the tree is 30°, you will be standing at half the tree's height (where the sine values apply). A triangle made from you to the top of the tree forms similar ratios that reflect these trigonometric values.

Trigonometric Ratios of 0° and 90°

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Let us see what happens to the trigonometric ratios of angle A, if it is made smaller and smaller in the right triangle ABC (see Fig. 8.16), till it becomes zero. As ∠ A gets smaller and smaller, the length of the side BC decreases. The point C gets closer to point B, and finally when ∠ A becomes very close to 0°, AC becomes almost the same as AB (see Fig. 8.17).

When ∠ A is very close to 0°, BC gets very close to 0 and so the value of sin A = BC/AC is very close to 0. Also, when ∠ A is very close to 0°, AC is nearly the same as AB and so the value of cos A = AC/AB is very close to 1. This helps us to see how we can define the values of sin A and cos A when A = 0°.

We define: sin 0° = 0 and cos 0° = 1. Using these, we have :
- tan 0° = sin 0° / cos 0° = 0,
- cot 0° is not defined,
- sec 0° = 1/cos 0° = 1,
- cosec 0° is not defined.

Detailed Explanation

As angle A approaches 0°, we intuitively understand that the height from the base converges to a line, leading to near-zero sine while the cosine ratio remains close to 1. These limits help us differenciate between values of these trigonometric functions at boundaries. At 0°, we can establish known values: sin 0°=0 denotes the absence of height, while cos addresses the full horizontal stretch.

Examples & Analogies

Picture diving from a diving board. As you lean forward to dive, you angle towards the water (θ approach 0°). Ultimately, when you dive straight, you're close to the horizontal which represents cos (like being flat) and your height drops to zero, portrayed as sin (falling into the water).

Definitions & Key Concepts

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

Key Concepts

  • Trigonometric Ratios: Defined for angles using sides of right triangles.

  • Sine of an Angle: Ratio of opposite side to hypotenuse.

  • Cosine of an Angle: Ratio of adjacent side to hypotenuse.

  • Values for Special Angles: Specific sine, cosine, and tangent values for 0°, 30°, 45°, 60°, and 90°.

Examples & Real-Life Applications

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

Examples

  • Finding sin 30°: By constructing a right triangle, sin 30° = 1/2.

  • Calculating tan 45°: Using the definition, tan 45° = 1 since both opposing sides are equal.

Memory Aids

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

🎵 Rhymes Time

  • At 30° and 60° in trigonometry's dance, sin and cos take their chance.

📖 Fascinating Stories

  • Imagine a triangle where an angle shines bright at 45°. The pairs of sides represent equal delight!

🧠 Other Memory Gems

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

🎯 Super Acronyms

For 45° remember SCC

  • Sine
  • Cosine
  • Correspond equally.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Trigonometric Ratios

    Definition:

    Relational ratios of the sides of a right triangle, defining the sine, cosine, tangent, cosecant, secant, and cotangent functions.

  • Term: Sine

    Definition:

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

  • Term: Cosine

    Definition:

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

  • Term: Tangent

    Definition:

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

  • Term: Cosecant

    Definition:

    Reciprocal of sine, or 1/sin.

  • Term: Secant

    Definition:

    Reciprocal of cosine, or 1/cos.

  • Term: Cotangent

    Definition:

    Reciprocal of tangent, or 1/tan.