Cofunction Identities Examples & Practice Problems Trigonometry

The Organic Chemistry Tutor
15 Nov 201705:42
EducationalLearning
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TLDRThe video script focuses on cofunction identities, which are mathematical relationships between trigonometric functions of complementary angles. The presenter explains that sine and cosine, tangent and cotangent, secant and cosecant are cofunctions, meaning they are equal when their respective angles add up to 90 degrees. Several examples are given to illustrate this concept, such as sine 30 degrees being equal to cosine 60 degrees, and sine 20 degrees being equal to cosine 70 degrees. The script also provides practice problems for the viewer to identify the cofunctions of given angles, like sine 65 degrees being equivalent to cosine 25 degrees, and tangent pi/3 being equivalent to tangent pi/6. The video concludes by reinforcing the understanding of cofunction identities and their practical applications in trigonometry.

Takeaways
  • πŸ“š The video focuses on explaining cofunction identities, which are mathematical relationships between trigonometric functions of complementary angles.
  • πŸ“ The cofunction identities presented are: sine theta equals cosine of 90 degrees minus theta, and cosine theta equals sine of 90 degrees minus theta.
  • πŸ“‰ Tangent is the cofunction of cotangent, and vice versa, with the formula being tangent theta equals cotangent of 90 degrees minus theta.
  • πŸ“ˆ Secant and cosecant are also cofunctions, where secant theta equals cosecant of 90 degrees minus theta, and cosecant theta equals secant of 90 degrees minus theta.
  • πŸ”’ The video provides examples to illustrate the concept, such as sine 30 degrees being equal to cosine 60 degrees, both of which equal 1/2.
  • πŸ‘‰ It is emphasized that sine and cosine are equal when the sum of their angles is 90 degrees, showcasing the complementary nature of these trigonometric functions.
  • πŸ“ Practice problems are given to help viewers identify the cofunctions with appropriate angles, such as sine 65 degrees being equal to cosine 25 degrees.
  • πŸ“² The video suggests using a calculator in degree mode to verify the results of the cofunction identities.
  • πŸ“š The importance of converting angles to a common unit (degrees or radians) is highlighted, especially when dealing with pi (Ο€) as in the example of cotangent pi/3.
  • πŸ“‰ The video demonstrates how to find the cofunction of secant 42 degrees by subtracting the angle from 90 degrees, resulting in cosecant 48 degrees.
  • πŸ“ˆ The final example involves finding the cofunction of cosine pi/8, which is done by subtracting the angle from pi/2, resulting in sine 3pi/8.
Q & A

  • What is the relationship between sine and cosine functions as described in the video?

    -The relationship between sine and cosine functions is that sine theta is equal to cosine of (90 - theta) and vice versa. This means that when the sum of two angles is 90 degrees, the sine of one angle is equal to the cosine of the other.

  • What is the formula for the cofunction identity of tangent and cotangent?

    -The cofunction identity for tangent and cotangent is that tangent is equal to cotangent of (90 - theta) and cotangent is equal to tangent of (90 - theta).

  • How does the secant function relate to the cosecant function in the context of cofunction identities?

    -In the context of cofunction identities, secant is equal to cosecant of (90 - theta) and cosecant is equal to secant of (90 - theta).

  • What is the value of sine 30 degrees in terms of cosine function?

    -The value of sine 30 degrees is equal to the cosine of (90 - 30) degrees, which is cosine 60 degrees. Both sine 30 and cosine 60 are equal to 1/2.

  • If sine of an angle is equal to cosine of another angle, what is the sum of those two angles?

    -If the sine of an angle is equal to the cosine of another angle, the sum of those two angles is 90 degrees.

  • What is the cofunction of sine 20 degrees?

    -The cofunction of sine 20 degrees is cosine 70 degrees, because 20 degrees plus 70 degrees equals 90 degrees.

  • What is the cofunction of cotangent pi/3 in radians?

    -The cofunction of cotangent pi/3 is tangent (pi/2 - pi/3), which simplifies to tangent pi/6 in radians.

  • How is the cofunction identity of secant 42 degrees calculated?

    -The cofunction of secant 42 degrees is cosecant (90 - 42), which simplifies to cosecant 48 degrees.

  • What is the cofunction of cosine pi/8 in radians?

    -The cofunction of cosine pi/8 is sine (pi/2 - pi/8), which simplifies to sine (3pi/8) in radians.

  • Why is it important to ensure the calculator is in degree mode when calculating trigonometric functions?

    -It is important to ensure the calculator is in degree mode to avoid incorrect calculations, as the values of trigonometric functions differ significantly between degrees and radians.

  • How do cofunction identities help in solving trigonometric problems?

    -Cofunction identities help in solving trigonometric problems by providing a relationship between different trigonometric functions, allowing for easier conversion and calculation when one function's value is known.

  • What is the significance of the value of sine 0 and cosine 90 in the context of cofunction identities?

    -The significance of sine 0 and cosine 90 both being equal to 0 is that it demonstrates a special case where the sine of one angle and the cosine of its cofunction angle have the same value, highlighting the relationship between the two functions.

Outlines
00:00
πŸ“š Introduction to Cofunction Identities

The video begins with an introduction to cofunction identities in trigonometry. The presenter lists the key formulas: sine theta equals cosine of 90 degrees minus theta, and vice versa for cosine and sine. The same relationship applies to tangent and cotangent, as well as secant and cosecant. The presenter uses examples to illustrate how these identities work, such as sine 30 degrees being equal to cosine 60 degrees, both of which equal 1/2. The purpose of the video is to help viewers understand and apply these identities to solve problems involving trigonometric functions.

05:02
🧩 Applying Cofunction Identities to Practice Problems

The second paragraph continues with practical applications of cofunction identities. The presenter provides several problems for the viewers to solve, such as finding the cofunction of sine 65 degrees, which is cosine 25 degrees, since 90 degrees minus 65 degrees equals 25 degrees. Other examples include finding the cofunction of tangent 20 degrees (cotangent 70 degrees), cotangent pi/3 (tangent pi/6), secant 42 degrees (cosecant 48 degrees), and cosine pi/8 (sine 3pi/8). The presenter guides the viewers through the process of converting each angle and identifying the correct cofunction, demonstrating the use of cofunction identities in problem-solving.

Mindmap
Keywords
πŸ’‘Cofunction Identities
Cofunction Identities are mathematical relationships between trigonometric functions of complementary angles. In the context of the video, these identities are essential for understanding how the sine of an angle is equal to the cosine of its complement (90 degrees minus the angle), and vice versa for other trigonometric functions. The video script provides several examples to illustrate these identities, such as sine 30 degrees being equal to cosine 60 degrees, emphasizing their importance in solving trigonometric problems.
πŸ’‘Sine
Sine is a fundamental trigonometric function that relates the ratio of the opposite side to the hypotenuse in a right-angled triangle. The video explains that sine of an angle (ΞΈ) is equal to the cosine of its complement (90 degrees - ΞΈ). For instance, sine 30 degrees is equal to cosine 60 degrees, which simplifies to 1/2, demonstrating the application of cofunction identities.
πŸ’‘Cosine
Cosine, like sine, is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The video script highlights the cofunction identity of cosine, stating that the cosine of an angle is equal to the sine of its complement. This is exemplified when discussing sine 20 degrees, which is equal to cosine 70 degrees, showcasing the complementary relationship between sine and cosine.
πŸ’‘Tangent
Tangent is a trigonometric function defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. The video script explains the cofunction identity of tangent, which states that tangent of an angle is equal to the cotangent of its complement. An example given is tangent 20 degrees, which is the cofunction of cotangent 70 degrees.
πŸ’‘Cotangent
Cotangent is the reciprocal of the tangent function, and it is also a trigonometric function. The video script describes the cofunction identity of cotangent, which is that cotangent of an angle is equal to the tangent of its complement. This is illustrated with the example of cotangent Ο€/3 radians, which is the cofunction of tangent (Ο€/2 - Ο€/3) radians.
πŸ’‘Secant
Secant is the reciprocal of the cosine function and is used in trigonometry to describe the ratio of the hypotenuse to the adjacent side in a right-angled triangle. The video script mentions that secant of an angle is equal to the cosecant of its complement. An example provided is secant 42 degrees, which is the cofunction of cosecant (90 - 42) degrees.
πŸ’‘Cosecant
Cosecant is the reciprocal of the sine function and represents the ratio of the hypotenuse to the opposite side in a right-angled triangle. The video script explains that the cosecant of an angle is equal to the secant of its complement. This concept is demonstrated with the problem involving secant 42 degrees, which equals the cosecant of 48 degrees.
πŸ’‘Complementary Angles
Complementary angles are two angles that add up to 90 degrees. The video script uses the concept of complementary angles to explain cofunction identities, such as sine of an angle being equal to the cosine of its complement. This is crucial for understanding how the values of trigonometric functions relate to each other when dealing with angles that sum to 90 degrees.
πŸ’‘Radians
Radians are a unit of angular measure used in trigonometry, where a full circle is 2Ο€ radians. The video script mentions radians in the context of converting between degrees and radians when dealing with angles in trigonometric functions, such as when finding the cofunction of cotangent Ο€/3 radians by using the relationship with tangent (Ο€/2 - Ο€/3) radians.
πŸ’‘Practice Problems
Practice problems are exercises given at the end of the video script to help viewers apply and understand the concept of cofunction identities. The video provides specific angles and asks viewers to find their cofunctions, such as sine 65 degrees, tangent 20 degrees, and others, reinforcing the learning objective of the video.
πŸ’‘Trigonometric Functions
Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The video script focuses on several key trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, and their cofunction identities, which are essential for solving various trigonometric equations and problems.
Highlights

Introduction to cofunction identities in trigonometry.

Formula for sine theta as the cosine of 90 degrees minus theta.

Formula for cosine theta as the sine of 90 degrees minus theta.

Tangent being the cofunction of cotangent for the complementary angle.

Cotangent being the cofunction of tangent for the complementary angle.

Secant being the cofunction of cosecant for the complementary angle.

Cosecant being the cofunction of secant for the complementary angle.

Explanation of how sine and cosine are cofunctions that add up to 90 degrees.

Example calculation: sine 30 degrees equals cosine of 60 degrees.

Demonstration that sine and cosine of complementary angles sum to 1.

Example: sine 20 degrees equals cosine of 70 degrees.

Example: sine 10 degrees equals cosine of 80 degrees.

Special case: sine 0 equals cosine 90, both being 0.

Special case: sine 45 equals cosine 45, both being √2/2.

Practice problems to identify cofunctions with appropriate angles.

Solution for practice problem: sine 65 degrees equals cosine 25 degrees.

Solution for practice problem: tangent 20 degrees equals cotangent 70 degrees.

Conversion of angles from degrees to radians for cotangent pi/3.

Solution involving common denominators for cotangent pi/3.

Solution for secant 42 degrees as cosecant of 48 degrees.

Solution for cosine pi/8 as sine of 3 pi/8.

Conclusion summarizing the understanding of cofunction identities.

Transcripts

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