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Chapter 3: Problem 95

State the three Pythagorean identities.

Short Answer

Expert verified
1. \( \text{sin}^2(x) + \text{cos}^2(x) = 1 \) 2. \( 1 + \text{tan}^2(x) = \text{sec}^2(x) \) 3. \( \text{cot}^2(x) + 1 = \text{csc}^2(x) \)

Step by step solution

01

- Define the Primary Pythagorean Identity

The first and most fundamental Pythagorean identity is derived from the Pythagorean Theorem. It states \[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \]
02

- Derive the Second Pythagorean Identity

The second Pythagorean identity can be obtained by dividing the first identity by \( \text{cos}^2(x) \). This gives us \[ 1 + \text{tan}^2(x) = \text{sec}^2(x) \]
03

- Derive the Third Pythagorean Identity

The third Pythagorean identity can be derived by dividing the first identity by \( \text{sin}^2(x) \). This results in \[ \text{cot}^2(x) + 1 = \text{csc}^2(x) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Primary Pythagorean Identity
The **Primary Pythagorean Identity** is a fundamental concept in trigonometry. It originates from the Pythagorean Theorem, which applies to right-angled triangles. By considering the unit circle, where the hypotenuse is always 1, the identity comes out as follows: \[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \] This identity establishes a relationship between the sine and cosine of an angle. Being fundamental, it serves as the base for the other Pythagorean identities. Whether you are analyzing triangles, solving equations, or proving other identities, understanding this primary identity is crucial.
Trigonometric Identities
**Trigonometric identities** are equations that are true for all values of the variables involved. They include relationships between trigonometric functions, like sine, cosine, and tangent. The three Pythagorean identities are key examples:
  • **First Identity**: \[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \]
  • **Second Identity**: Dividing the first identity by \[ \text{cos}^2(x) \] yields \[ 1 + \text{tan}^2(x) = \text{sec}^2(x) \]
  • **Third Identity**: Dividing the first identity by \[ \text{sin}^2(x) \] gives us \[ \text{cot}^2(x) + 1 = \text{csc}^2(x) \]
These identities allow us to transform and simplify trigonometric expressions. They are essential tools for solving various mathematical and physical problems.
Trigonometry
**Trigonometry** is the branch of mathematics that studies relationships between side lengths and angles of triangles. At its core, trigonometry deals with trigonometric functions such as sine (sin), cosine (cos), and tangent (tan). These functions are derived from the unit circle and right-angled triangles.
Some important points about trigonometry:
  • **Unit Circle**: A circle with a radius of 1 centered at the origin. It helps define sine, cosine, and tangent for all angles.
  • **Right-Angled Triangles**: Triangles with one 90-degree angle. Trigonometric functions represent ratios of side lengths.
  • **Applications**: Trigonometry is used in various fields, including physics, engineering, astronomy, and architecture.
Understanding basic trigonometric identities and their derivations is vital. It forms a strong foundation for more advanced math and real-world problem-solving.

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Most popular questions from this chapter

Find the exact value of \(\cos (\alpha+\beta)\) if \(\sin \alpha=2 / 3\) and \(\sin \beta=-1 / 2,\) with \(\alpha\) in quadrant \(I\) and \(\beta\) in quadrant III.

Find the exact value of \(\cos (\alpha-\beta)\) if \(\cos \alpha=\sqrt{3} / 4\) and \(\cos \beta=-\sqrt{2} / 3,\) with \(\alpha\) in quadrant \(I\) and \(\beta\) in quadrant II.

Prove that each of the following equations is an identity. HINT \(\ln (a / b)=\ln (a)-\ln (b)\) and \(\ln (a b)=\ln (a)+\ln (b)\) for \(a>0\) and \(b>0\) $$ \ln |\sec \alpha+\tan \alpha|=-\ln |\sec \alpha-\tan \alpha| $$

Prove that each of the following equations is an identity. HINT \(\ln (a / b)=\ln (a)-\ln (b)\) and \(\ln (a b)=\ln (a)+\ln (b)\) for \(a>0\) and \(b>0\) $$ \ln |\csc \alpha+\cot \alpha|=-\ln |\csc \alpha-\cot \alpha| $$

52\. \(\frac{1-\cos ^{2}\left(\frac{x}{2}\right)}{1-\sin ^{2}\left(\frac{x}{2}\right)}=\frac{1-\cos x}{1+\cos x}\)
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