Pythagorean Identities - Examples & Practice Problems, Trigonometry

The Organic Chemistry Tutor
13 Oct 201707:43
EducationalLearning
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TLDRThe video script delves into the Pythagorean identities, focusing on their applications in trigonometry. It explains three key identities: sine squared plus cosine squared equals one, one plus tangent squared equals secant squared, and one plus cotangent squared equals cosecant squared. The script provides step-by-step examples to demonstrate how to find the value of cosine or sine when one is given, using the first identity. It also discusses the importance of considering the quadrant in which the angle lies to determine the correct sign for the trigonometric function. The examples cover scenarios where sine or cosine values are known, and the script guides viewers on how to calculate the missing trigonometric value while considering the angle's quadrant.

Takeaways
  • πŸ“š The script discusses three Pythagorean identities: sine squared plus cosine squared equals one, one plus tangent squared equals secant squared, and one plus cotangent squared equals cosecant squared.
  • πŸ” The first identity, sine squared theta plus cosine squared theta equals one, is the primary focus and is used to find the value of cosine when sine is known.
  • πŸ“‰ Given sine theta equals 4/5 and angle theta is between 0 and 90 degrees, the script demonstrates how to calculate cosine theta as positive 3/5 using the Pythagorean identity.
  • πŸ“ The script explains that the value of cosine is found by subtracting sine squared from one and then taking the square root of the result.
  • πŸ“ˆ The sign of the cosine value is determined by the quadrant in which the angle theta lies; in quadrant one, cosine is positive.
  • πŸ”„ Another example is given where cosine theta equals 8/17 and angle theta is between 3Ο€/2 and 2Ο€, indicating that the angle is in quadrant four where sine is negative and cosine is positive.
  • πŸ“ The script shows the calculation of sine theta when given cosine theta equals 8/17, resulting in sine theta being negative 15/17.
  • πŸ”’ A third example is presented where sine theta equals 2/5 and tangent theta is less than zero, indicating the angle is in quadrant four where both sine and cosine are positive.
  • πŸ“Œ The script uses the Pythagorean identity to find cosine theta as positive square root of 21 over 5 when sine theta is 2/5 and tangent theta is negative.
  • πŸ“˜ The importance of understanding the quadrant in which the angle lies is emphasized to determine the correct sign of the trigonometric values.
  • πŸ“š The script concludes by reinforcing the use of the Pythagorean identity to find the missing trigonometric function value based on the known function and angle quadrant.
Q & A

  • What are the three Pythagorean identities mentioned in the script?

    -The three Pythagorean identities mentioned are: 1) sine squared theta plus cosine squared theta equals one, 2) one plus tangent squared equals secant squared, and 3) one plus cotangent squared equals cosecant squared.

  • If sine theta equals 4/5 and theta is between 0 and 90 degrees, how do you find the value of cosine theta?

    -Using the Pythagorean identity sine squared plus cosine squared equals one, and knowing sine theta is 4/5, you square both sides (16/25) and subtract from 1 (25/25) to get cosine squared theta equals 9/25. Taking the square root gives cosine theta as Β±3/5. Since theta is in quadrant one (0 to 90 degrees), where cosine is positive, the value is +3/5.

  • What is the significance of the quadrant in determining the sign of the trigonometric functions?

    -The quadrant in which the angle theta lies determines the signs of the trigonometric functions. For example, in quadrant one, both sine and cosine are positive, while in quadrant four, sine is negative and cosine is positive.

  • How does the script use the Pythagorean identity to find the value of sine theta when cosine theta is given as 8/17 and theta is between 3Ο€/2 and 2Ο€?

    -The script uses the identity sine squared plus cosine squared equals one to find sine. With cosine theta as 8/17, squaring both sides gives 64/289. Subtracting this from 1 (289/289) gives sine squared theta as 225/289. Taking the square root gives sine theta as Β±15/17. Since theta is in quadrant four (between 270 and 360 degrees), where sine is negative, the value is -15/17.

  • What is the relationship between tangent and the quadrants in the script's third example?

    -In the third example, the script explains that tangent is less than zero in quadrants two and four. Since sine is positive and tangent is negative, the angle must be in quadrant four where cosine is positive.

  • How does the script determine the value of cosine theta when sine theta is 2/5 and tangent theta is less than zero?

    -Using the identity sine squared plus cosine squared equals one, and knowing sine theta is 2/5, the script calculates cosine squared theta as 21/25. Taking the square root gives cosine theta as ±√21/5. Since the angle is in quadrant four where cosine is positive, the value is +√21/5.

  • Why is it necessary to consider the quadrant when finding the trigonometric function values?

    -Considering the quadrant is necessary because it determines the signs of the trigonometric functions. Without knowing the quadrant, you cannot determine whether the functions should be positive or negative.

  • What is the formula used to find the value of cosine when sine is given?

    -The formula used to find the value of cosine when sine is given is cosine squared theta equals one minus sine squared theta, derived from the Pythagorean identity sine squared plus cosine squared equals one.

  • How does the script handle the square root of a fraction in the process of finding cosine or sine values?

    -The script separates the square root of the numerator and the square root of the denominator when handling the square root of a fraction, as seen when finding cosine theta with sine theta as 2/5.

  • Can the script's method be used to find the value of tangent or cotangent using the Pythagorean identities?

    -While the script focuses on sine and cosine, the method of using Pythagorean identities can be adapted to find tangent or cotangent values by using the identities one plus tangent squared equals secant squared and one plus cotangent squared equals cosecant squared.

  • What is the significance of the angle theta being between 0 and 90 degrees in the first example?

    -The significance is that it places theta in quadrant one, where both sine and cosine are positive. This information is crucial for determining the correct sign of the cosine value.

Outlines
00:00
πŸ“š Pythagorean Identities and Solving for Cosine Theta

This paragraph introduces three Pythagorean identities essential for trigonometry: sine squared plus cosine squared equals one, one plus tangent squared equals secant squared, and one plus cotangent squared equals cosecant squared. The focus is on the first identity and its application to find the value of cosine theta when sine theta is given. An example is provided where sine theta equals 4/5 and angle theta is between 0 and 90 degrees. Using the identity, the calculation leads to cosine theta being positive 3/5, considering the angle's location in the first quadrant where cosine is positive.

05:01
πŸ” Determining Sine Theta and Cosine Theta Using Quadrants

The second paragraph explores the use of trigonometric identities to determine the values of sine and cosine for given angles, with a focus on understanding which quadrant the angle lies in to choose the correct sign for the values. Two examples are given: one where cosine theta is 8/17 and the angle is between 3Ο€/2 and 2Ο€, indicating quadrant 4 where sine is negative, leading to sine theta being -15/17. The other example involves sine theta being 2/5 with tangent theta less than zero, which places the angle in quadrant 4. The calculation for cosine theta results in a positive value of √21/5, as cosine is positive in quadrant 4.

Mindmap
Keywords
πŸ’‘Pythagorean Identities
Pythagorean Identities are fundamental mathematical relationships in trigonometry that relate the squares of the sine, cosine, tangent, secant, cotangent, and cosecant functions of an angle. In the video, three specific identities are discussed: sine squared plus cosine squared equals one, one plus tangent squared equals secant squared, and one plus cotangent squared equals cosecant squared. These identities are crucial for solving trigonometric problems and are central to the theme of the video, which is to demonstrate how to use these identities to find the values of trigonometric functions.
πŸ’‘sine squared
Sine squared, denoted as \(\sin^2(\theta)\), is the square of the sine function of an angle theta. In the context of the video, sine squared is used in the first Pythagorean identity, where it is added to cosine squared to equal one (\(\sin^2(\theta) + \cos^2(\theta) = 1\)). This relationship is key to finding the cosine of an angle when the sine is known, as illustrated in the first example where sine theta equals 4/5.
πŸ’‘cosine squared
Cosine squared, represented as \(\cos^2(\theta)\), is the square of the cosine function of an angle theta. It is used in the video to find the value of cosine when sine is given, by rearranging the first Pythagorean identity. For instance, when sine theta is 4/5, the script calculates cosine by subtracting sine squared from one and then taking the square root, resulting in cosine theta being 3/5.
πŸ’‘tangent squared
Tangent squared, written as \(\tan^2(\theta)\), is the square of the tangent function. It is part of the second Pythagorean identity mentioned in the video, which states that one plus tangent squared equals secant squared (\(1 + \tan^2(\theta) = \sec^2(\theta)\)). Although not used in the examples, tangent squared is an important concept in trigonometry for understanding the relationship between the angle's tangent and secant.
πŸ’‘secant squared
Secant squared, denoted as \(\sec^2(\theta)\), is the reciprocal of the square of the cosine function. It is introduced in the video as part of the second Pythagorean identity. While not directly used in the examples, secant squared is a significant concept that helps in understanding the relationship between the angle's secant and tangent.
πŸ’‘cotangent squared
Cotangent squared, represented as \(\cot^2(\theta)\), is the square of the cotangent function. It is part of the third Pythagorean identity, which states that one plus cotangent squared equals cosecant squared (\(1 + \cot^2(\theta) = \csc^2(\theta)\)). Similar to secant squared, cotangent squared is mentioned but not used in the examples, yet it is essential for understanding trigonometric relationships.
πŸ’‘Quadrant
In the context of the video, a quadrant refers to one of the four equal divisions of a coordinate plane by the two axes. Quadrants are important for determining the signs of trigonometric functions. For example, the video explains that in quadrant one (0 to 90 degrees), both sine and cosine are positive, while in quadrant four (270 to 360 degrees or 3\pi/2 to 2\pi radians), sine is negative and cosine is positive.
πŸ’‘Square root
The square root operation is used in the video to find the cosine of an angle when dealing with the Pythagorean identity. After calculating the value of cosine squared, the square root is taken to find the actual cosine value. For example, when cosine squared is 9/25, the square root of both the numerator and the denominator gives the cosine value of 3/5.
πŸ’‘Trigonometric functions
Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the video, sine, cosine, tangent, secant, cotangent, and cosecant are all trigonometric functions that are used to solve for unknown angles or sides in right-angled triangles. The video's theme revolves around using these functions and their relationships to find missing values.
πŸ’‘Radians
Radians are a unit of angular measure used in trigonometry. While not explicitly mentioned in the script, the concept of radians is implied when discussing angles in trigonometric functions. For example, the video mentions '3 pi over 2', which is a radian measure equivalent to 270 degrees. Understanding radians is important for accurately working with trigonometric functions, especially in calculus and more advanced mathematics.
πŸ’‘Degrees
Degrees are a unit of angular measurement used in the video to express angles. The script specifies that the angle theta is between 0 and 90 degrees in the first example and between 270 and 360 degrees in the second example. Degrees are essential for understanding the context of the examples provided and for determining the signs of trigonometric functions in different quadrants.
Highlights

Introduction to the Pythagorean identities in trigonometry.

Three main Pythagorean identities are sine squared plus cosine squared equals one, one plus tangent squared equals secant squared, and one plus cotangent squared equals cosecant squared.

Using the first Pythagorean identity to find the value of cosine when sine is given.

Example calculation where sine theta equals 4/5 and angle theta is between 0 and 90 degrees.

Demonstration of finding cosine theta using the identity sine squared plus cosine squared equals one.

Determining the sign of cosine based on the quadrant in which the angle lies.

Another example where cosine theta equals 8/17 and angle theta is between 3pi/2 and 2pi.

Using the Pythagorean identity to find sine theta when cosine theta is given.

Determining the sign of sine based on the quadrant of the angle.

Example with sine theta equals 2/5 and tangent theta is less than zero.

Identifying the quadrant of the angle based on the signs of sine and tangent.

Finding the value of cosine theta when sine theta is given and the angle is in a specific quadrant.

Explanation of the process to find cosine theta using the Pythagorean identity and choosing the correct sign based on the quadrant.

Final answer for cosine theta when the angle is in quadrant four and sine is positive but tangent is negative.

Emphasis on the importance of the quadrant in determining the signs of trigonometric functions.

Transcripts

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