Even and Odd Trigonometric Functions & Identities - Evaluating Sine, Cosine, & Tangent

The Organic Chemistry Tutor
24 Jun 201710:51
EducationalLearning
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TLDRThe video discusses even and odd trigonometric functions, focusing on cosine and secant as even functions, and sine, cosecant, tangent, and cotangent as odd functions. It explains how to apply this knowledge to find values of trigonometric functions at specific angles, using examples such as cosine of -45 degrees, sine of pi/3, tangent of -3pi/4, cosecant of 5pi/6, and secant of -11pi/6. The video emphasizes the properties of these functions in different quadrants of the unit circle to determine positive or negative values.

Takeaways
  • 📚 Cosine and secant are identified as even functions, meaning \( \cos(-x) = \cos(x) \) and \( \sec(-x) = \sec(x) \).
  • 🔍 Sine, tangent, and cosecant are classified as odd functions, with properties \( \sin(-x) = -\sin(x) \), \( \tan(-x) = -\tan(x) \), and \( \csc(-x) = -\csc(x) \).
  • 📈 The script demonstrates how to use the even and odd properties to find the values of trigonometric functions at negative angles.
  • 📐 For cosine of -45°, the script explains that it is equivalent to the positive cosine of 45°, which is \( \frac{\sqrt{2}}{2} \).
  • 📉 The script shows that sine of \( \frac{\pi}{3} \) is negative \( \frac{\sqrt{3}}{2} \), considering the sign conventions in the fourth quadrant.
  • 🤔 The tangent of \( -\frac{3\pi}{4} \) is calculated as positive 1, illustrating the odd function property of tangent.
  • 🔢 The cosecant of \( -\frac{5\pi}{6} \) is found to be negative, following the odd function property and considering the signs in the third quadrant.
  • 🔄 The script explains that secant, being the reciprocal of cosine, is an even function, and thus \( \sec(-x) = \sec(x) \).
  • 📌 The value of secant at \( -\frac{11\pi}{6} \) is determined to be positive, using the reciprocal identity and considering the signs in the fourth quadrant.
  • 🧩 The script concludes with the value of tangent at \( \pi \), which is zero, as both sine and cosine are zero at this angle, resulting in an undefined division.
Q & A

  • What are the characteristics of an even function in trigonometry?

    -An even function in trigonometry is symmetric with respect to the y-axis, meaning that f(x) = f(-x). Examples from the script include cosine and secant functions, which maintain the same value for both positive and negative input angles.

  • What are the characteristics of an odd function in trigonometry?

    -An odd function in trigonometry is symmetric with respect to the origin, meaning that f(x) = -f(-x). Examples from the script include sine, cosecant, tangent, and cotangent functions, which change sign when the input angle is negated.

  • Why is the cosine function considered an even function?

    -The cosine function is considered even because it satisfies the condition f(x) = f(-x). This is demonstrated in the script by showing that cosine of an angle x is equal to the cosine of the negative of that angle (cos(x) = cos(-x)).

  • How does the secant function relate to the cosine function in terms of evenness?

    -The secant function, being the reciprocal of the cosine function, also exhibits even function characteristics. This is because secant of an angle x is equal to secant of the negative of that angle (sec(x) = sec(-x)), as explained in the script.

  • What is the significance of the sine function being an odd function?

    -The sine function being an odd function signifies that it satisfies the condition f(x) = -f(-x). This means that sine of an angle x is equal to the negative of sine of the negative of that angle (sin(x) = -sin(-x)), as illustrated in the script.

  • How does the tangent function behave when given a negative angle?

    -As an odd function, the tangent of a negative angle is the negative of the tangent of the positive angle. For example, tangent of -x is equal to negative tangent of x (tan(-x) = -tan(x)), which is explained in the script.

  • What is the value of cosine at -45° based on the even function property?

    -Using the property of cosine being an even function, the value of cosine at -45° is equivalent to the value of cosine at 45°. The script states that cosine -45° is the same as cosine 45°, which corresponds to the point on the unit circle (√2/2, √2/2).

  • How does the sine function behave at an angle of 3π/4?

    -The sine function, being an odd function, will have a negative value at an angle of 3π/4 compared to π/4. The script explains that sin(π/3) is positive √3/2, but sin(3π/4) will be negative √3/2 because it is in the fourth quadrant where sine is negative.

  • What is the value of tangent at -3π/4, and how does it demonstrate the odd function property?

    -The script explains that tangent at -3π/4 is equal to the positive value of tangent at 3π/4 because tangent is an odd function. At 3π/4, the tangent is calculated as y/x, which results in a positive 1 because both x and y are negative in the third quadrant, and dividing a negative by a negative yields a positive.

  • How does the cosecant function behave at an angle of 5π/6?

    -The cosecant function, being the reciprocal of the sine function and thus an odd function, will have a negative value at an angle of 5π/6 compared to π/6. The script shows that cosecant 5π/6 is -2 because it is in the second quadrant where x is negative and y is positive, leading to a negative result when taking the reciprocal of sine.

  • What is the value of secant at -11π/6, and how does it demonstrate the even function property?

    -The secant function, being an even function, will have the same value at an angle of -11π/6 as it does at 11π/6. The script demonstrates that secant 11π/6 is positive because it is in the fourth quadrant where cosine, and thus secant, is positive. The exact value is 2√3/3 after rationalizing the denominator.

  • What is the value of tangent at π, and how does it demonstrate the odd function property?

    -The script concludes with the evaluation of tangent at π, which is zero. This is because tangent π is the same as negative tangent of -π, and since both π and -π lead to the same point on the unit circle (-1, 0), the tangent (y/x) results in zero because the denominator is zero.

Outlines
00:00
📚 Understanding Even and Odd Trigonometric Functions

This paragraph introduces the concepts of even and odd functions within the context of trigonometry. It explains that cosine and secant are even functions, meaning their values remain the same regardless of the sign of the input angle (e.g., cos(x) = cos(-x)). On the other hand, sine, cosecant, tangent, and cotangent are classified as odd functions, which implies that the sign of the output is determined by the sign of the input (e.g., sin(-x) = -sin(x)). The paragraph also demonstrates how to apply these properties to find the values of trigonometric functions at specific angles, using the unit circle and the concept of reference angles. Examples given include calculating cos(-45°), sin(π/3), and tan(-3π/4), illustrating the behavior of even and odd functions in different quadrants.

05:01
🔍 Analyzing the Properties of Tangent and Cosecant Functions

This section delves deeper into the properties of tangent and cosecant functions, emphasizing their odd nature. It explains that the tangent function, being odd, will have a negative value when the angle is negative, as demonstrated with tan(-3π/4) which equals -tan(3π/4). The explanation includes a visual representation of the unit circle to show how the signs of x and y coordinates affect the tangent value in different quadrants. The paragraph also covers the cosecant function, which is the reciprocal of the sine function and therefore also odd. Examples are provided to show how to find the value of cosecant at specific angles, such as -cosec(5π/6), by considering the reference angle and the signs of the coordinates in the respective quadrants.

10:07
📉 Evaluating Secant and Tangent at Specific Angles

The final paragraph discusses the evaluation of secant and tangent functions at particular angles, focusing on their even and odd properties, respectively. It starts with the secant function, which is an even function, and explains how to find sec(-11π/6) by considering the positive counterpart, sec(11π/6), due to the even nature of secant. The explanation includes finding the cosine value at π/6 and then taking its reciprocal to find the secant value. The paragraph concludes with an exploration of the tangent function at π, demonstrating that tan(π) equals zero because the tangent of π (or -π) results in a division by zero scenario, leading to an undefined value, which is then clarified as zero in the context of the unit circle.

Mindmap
Keywords
💡Even function
An even function is a mathematical function that satisfies the condition f(x) = f(-x) for all x in the function's domain. In the context of the video, cosine and secant are identified as even functions because their values remain the same when the input is negated. For instance, the script mentions that 'cosine of x is equal to cosine of -x', illustrating the property of evenness.
💡Odd function
An odd function is one that satisfies the condition f(x) = -f(-x) for all x in its domain. The video explains that sine, cosecant, tangent, and cotangent are odd functions. This means that the value of the function at a negative angle is the negative of its value at the corresponding positive angle, as demonstrated with 'sin(-x) = -sin(x)' and similar examples.
💡Cosine
Cosine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. In the video, cosine is used as an example of an even function, and its value at -45° is shown to be equivalent to its value at 45°, highlighting the symmetry inherent in cosine's behavior.
💡Secant
Secant is the reciprocal of the cosine function, denoted as sec(x) = 1/cos(x). It is used to find the angle when the ratio of the hypotenuse to the adjacent side of a right triangle is known. The video script states that 'secant of x is equal to secant x', emphasizing its even nature.
💡Sine
Sine is another basic trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right triangle. It is an odd function, as the video demonstrates with the example 'sin(-x) = -sin(x)'. The script also uses sine to explain the value at π/3, showing its odd property.
💡Cosecant
Cosecant, the reciprocal of the sine function (csc(x) = 1/sin(x)), is used to find the angle when the ratio of the hypotenuse to the opposite side is known. The video identifies cosecant as an odd function, and it provides an example with 'csc(5π/6)' to illustrate this property.
💡Tangent
Tangent is the ratio of the opposite side to the adjacent side in a right triangle, and it is represented as tan(x) = sin(x)/cos(x). The video script explains that tangent is an odd function, which is shown through the calculation of 'tan(-3π/4)', resulting in a positive value due to the division of two negative numbers.
💡Unit circle
The unit circle is a circle with a radius of 1 and is centered at the origin of a coordinate system. It is used in trigonometry to define the sine and cosine of an angle. The video uses the unit circle to determine the values of cosine and sine at specific angles, such as 45° and π/3, respectively.
💡Quadrant
In the context of the unit circle and coordinate planes, a quadrant refers to one of the four equal divisions. The video script discusses how the signs of trigonometric functions change depending on the quadrant in which the angle lies, as seen when explaining the values of sine and tangent at different angles.
💡Reciprocal identity
A reciprocal identity in trigonometry is a relationship where one function is the reciprocal of another. For example, secant is the reciprocal of cosine (sec(x) = 1/cos(x)), and cosecant is the reciprocal of sine (csc(x) = 1/sin(x)). The video uses these identities to find the values of secant and cosecant at specific angles.
💡Rationalize
Rationalizing a denominator is the process of eliminating radicals or other irrational numbers from the denominator of a fraction. In the video, the term is used when calculating sec(11π/6), where the result involves a square root in the denominator, and the process of rationalization is implied to reach the final answer.
Highlights

Cosine is an even function: cosine(x) = cosine(x).

Secant, the reciprocal of cosine, is also an even function: secant(x) = secant(x).

Sine is an odd function: sine(-x) = -sine(x).

Cosecant, the reciprocal of sine, is an odd function: cosecant(-x) = -cosecant(x).

Tangent is an odd function: tangent(-x) = -tangent(x).

Cotangent, the reciprocal of tangent, is an odd function: cotangent(-x) = -cotangent(x).

Cosine of -45 degrees is equivalent to cosine of 45 degrees.

Using the unit circle, cosine of 45 degrees is sqrt(2)/2.

Sine of pi/3 is sqrt(3)/2, but sine of -pi/3 is -sqrt(3)/2.

Tangent of -3pi/4 equals -1 due to it being in quadrant 3 where both x and y are negative.

Cosecant of 5pi/6 is -2 because it lies in quadrant 3 where y-values are negative.

Secant of -11pi/6 equals secant of 11pi/6, simplifying to 2/sqrt(3).

Using reciprocal identity, secant of 11pi/6 is rationalized to 2sqrt(3)/3.

Tangent of pi is zero because y/x at pi equals zero.

Positive pi and negative pi lead to the same point on the unit circle: (-1, 0).

Transcripts

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TrigonometryEven FunctionsOdd FunctionsUnit CircleCosineSineTangentSecantCosecantMath Tutorial