How To Evaluate Trigonometric Functions Using Periodic Properties - Trigonometry

The Organic Chemistry Tutor
13 Oct 201704:47
EducationalLearning
32 Likes 10 Comments

TLDRThe video script discusses the periodic properties of trigonometric functions, focusing on sine, cosine, tangent, and cotangent. It explains that adding 360 degrees or 2π to the angle of these functions results in the same value, a characteristic known as periodicity. The script uses examples to illustrate this concept, such as sine of 30 degrees being equal to sine of 390 degrees due to their coterminal relationship. It also demonstrates how to evaluate trigonometric functions at specific angles by finding coterminal angles less than 360 degrees, using sine and cosine examples at 420 degrees, -2π/3, and 750 degrees. The unit circle is referenced to determine the values of these functions, showing that the sine of 60 degrees and its coterminal angles like 420 and 750 are all equal to √3/2, and cosine of 4π/3 and -2π/3 is -1/2.

Takeaways
  • 🔁 The periodic property of trigonometric functions states that the sine of an angle is equal to the sine of the same angle plus 360 degrees or 2π.
  • 🔄 Coterminal angles, which differ by a multiple of 360 degrees, share the same trigonometric function values.
  • 📚 This periodic nature applies to all trigonometric functions, including sine, cosine, tangent, and cotangent.
  • 📉 To find a coterminal angle, you add or subtract multiples of 360 degrees from the original angle.
  • 📈 The graph of sine and cosine functions demonstrates their periodicity, repeating their values as the angle increases by multiples of 360 degrees.
  • 🧩 Evaluating trigonometric functions at specific angles can be simplified by finding a coterminal angle within the first 360 degrees.
  • 📌 Example given: Sine of 420 degrees is equal to sine of 60 degrees, which is \( \frac{\sqrt{3}}{2} \) based on the unit circle.
  • ➡️ Reference angles are used to find the values of cosine for angles outside the first 360 degrees, such as cosine of -2π/3 being equal to cosine of 4π/3.
  • 📐 At a unit circle, cosine is equal to the x-value of the point corresponding to the angle, which helps in determining the sign of the cosine value.
  • 🔢 Another example: Sine of 750 degrees simplifies to sine of 30 degrees, which is 1/2, due to the periodic nature of the sine function.
Q & A

  • What does the periodic property of trigonometric functions mean?

    -The periodic property of trigonometric functions means that the functions repeat their values in regular intervals. For sine, cosine, tangent, cotangent, etc., adding or subtracting multiples of 360 degrees (or 2π radians) to the angle results in the same function value.

  • How does the periodic property apply to the sine function?

    -The periodic property applies to the sine function such that sine of theta is equal to sine of theta plus any multiple of 360 degrees. This means sine of an angle and sine of its coterminal angle will have the same value.

  • What is a coterminal angle?

    -A coterminal angle is an angle that differs from the original angle by a full rotation, which is 360 degrees. Coterminal angles have the same trigonometric function values as their corresponding angles.

  • How can you find a coterminal angle for a given angle?

    -To find a coterminal angle for a given angle, you add or subtract multiples of 360 degrees to the original angle.

  • What is the value of sine 30 degrees and how does it relate to sine 390 degrees?

    -The value of sine 30 degrees is 1/2. Since sine 390 degrees is a coterminal angle of sine 30 degrees (390 - 360 = 30), sine 390 degrees also has the same value of 1/2.

  • How does the periodic property apply to the cosine function?

    -The periodic property applies to the cosine function such that cosine of an angle is equal to cosine of that angle plus any multiple of 360 degrees. This ensures that cosine of an angle and its coterminal angle will have the same value.

  • What is the value of cosine 50 degrees and how does it relate to cosine 410 degrees?

    -Cosine 50 degrees is equal to cosine 50 degrees plus 360 degrees, which is cosine 410 degrees. Therefore, both have the same value.

  • How can you use the periodic property to evaluate sine of 420 degrees?

    -To evaluate sine of 420 degrees, you find a coterminal angle that is less than 420 degrees by subtracting 360 degrees from 420 degrees, resulting in 60 degrees. Therefore, sine 420 degrees is equal to sine 60 degrees, which is root 3/2.

  • What is the value of cosine negative 2 PI/3 and how can you find it using the periodic property?

    -Cosine negative 2 PI/3 can be found by adding 2 PI to the angle, which gives cosine negative 2 PI/3 plus 2 PI, or cosine 4 PI/3. Since 4 PI/3 has the same reference angle as PI/3 (60 degrees), cosine 4 PI/3 is negative 1/2, which is the same as cosine negative 2 PI/3.

  • How can you evaluate sine of 750 degrees using the periodic property?

    -To evaluate sine of 750 degrees, you reduce the angle by subtracting multiples of 360 degrees until you get an angle less than 360. Subtracting 360 twice from 750 gives 30 degrees. Therefore, sine 750 degrees is equivalent to sine 30 degrees, which is 1/2.

  • What is the significance of the unit circle in evaluating trigonometric function values?

    -The unit circle is a circle with a radius of 1, where the angles are measured from the positive x-axis. It is significant in evaluating trigonometric function values because it provides a visual representation of the sine and cosine values for standard angles, making it easier to determine the values for coterminal angles.

Outlines
00:00
📚 Introduction to Periodic Properties of Trigonometric Functions

This paragraph introduces the concept of periodicity in trigonometric functions, specifically focusing on the sine function. It explains that sine of an angle (theta) is equal to sine of the same angle plus any multiple of 360 degrees or 2π. This property implies that sine values remain constant for coterminal angles, which are angles that differ by full rotations (360 degrees). The paragraph also mentions that this periodic property applies to other trigonometric functions such as cosine, tangent, and cotangent. The concept is illustrated with examples, showing how to find the sine value for angles like 30 degrees and 390 degrees, which are coterminal and thus share the same sine value. The explanation transitions into a discussion of how these functions are graphed, with sine and cosine functions repeating their patterns as the angle increases by multiples of 360 degrees.

Mindmap
Keywords
💡Trigonometric functions
Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the context of the video, these functions are used to describe periodic properties and are fundamental to understanding the behavior of sine, cosine, and tangent functions. The script discusses how these functions repeat their values for different angles, emphasizing their periodic nature.
💡Periodic properties
Periodic properties refer to the characteristic of a function to repeat its values at regular intervals. In the video, this concept is central as it explains how trigonometric functions like sine, cosine, and tangent repeat their values when the angle is increased or decreased by a full cycle, such as 360 degrees or 2π radians. The script uses this property to simplify the evaluation of trigonometric functions at different angles.
💡Coterminal angles
Coterminal angles are angles that share the same terminal side in standard position. The video explains that sine of an angle is equal to sine of that angle plus any multiple of 360 degrees, which means that angles differing by full rotations (360 degrees) are coterminal. This concept is used to find equivalent angles that are easier to work with, as their trigonometric function values are the same.
💡Unit circle
A unit circle is a circle with a radius of one, centered at the origin of a coordinate system. It is used in the context of the video to determine the values of trigonometric functions for specific angles. The script mentions the unit circle when explaining how to find the sine of 60 degrees, which corresponds to a point on the unit circle with specific x and y coordinates.
💡Reference angle
The reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. In the video, the reference angle is used to find the cosine of an angle greater than 90 degrees, such as 4π/3 radians. The script explains that the reference angle for 4π/3 is π/3, and the cosine value is determined based on the x-coordinate of the point on the unit circle corresponding to the reference angle.
💡Sine function
The sine function is one of the primary trigonometric functions that relates the ratio of the opposite side to the hypotenuse in a right-angled triangle. The video script uses the sine function to illustrate periodic properties, showing that sine of an angle is equal to sine of that angle plus or minus multiples of 360 degrees.
💡Cosine function
The cosine function is another fundamental trigonometric function that relates the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The video demonstrates the periodic nature of the cosine function, showing that cosine of an angle is equal to cosine of that angle plus or minus multiples of 360 degrees.
💡Tangent function
The tangent function is the ratio of the sine to the cosine of an angle. Although not explicitly detailed in the script, it is implied that the tangent function also exhibits periodic properties, as it is derived from the sine and cosine functions, which are both periodic.
💡Cotangent function
The cotangent function is the reciprocal of the tangent function, or the ratio of the adjacent side to the opposite side in a right-angled triangle. Similar to the tangent function, the cotangent function is periodic, as indicated in the video script, although it is not the primary focus of the explanation.
💡Evaluating trigonometric functions
Evaluating trigonometric functions involves finding the value of a trigonometric function for a given angle. The video script provides examples of how to evaluate sine and cosine functions at specific angles by using periodic properties and the unit circle. For instance, sine of 420 degrees is evaluated by finding a coterminal angle, 60 degrees, and using the known value from the unit circle.
Highlights

Sine of theta is equal to sine of theta plus 360 degrees or 2 pi.

Adding 360 degrees to theta results in the same sine value.

Coterminal angles share the same sine value.

To find a coterminal angle, add or subtract by 360 degrees.

Sine of 30 degrees and sine of 390 degrees have the same value.

This periodic property is true for all trigonometric functions: sine, cosine, tangent, cotangent.

Cosine of 50 degrees is equal to cosine of 50 degrees plus 360 degrees or cosine of 410 degrees.

The word 'periodic' means things that repeat.

Adding 2 pi to an angle results in the same cosine value.

When graphing sine and cosine functions, the pattern repeats over and over.

Evaluating sine of 420 degrees by finding a coterminal angle less than 420 degrees.

Sine of 420 degrees equals sine of 60 degrees.

Using the unit circle, sine of 60 degrees is root 3 over 2, so sine of 420 degrees is also root 3 over 2.

Cosine of negative 2 pi over 3 equals cosine of 4 pi over 3.

At 4 pi over 3, the x and y values are negative, so cosine 4 pi over 3 is negative 1/2.

Cosine of negative 2 pi over 3 is also negative 1/2.

Sine of 750 degrees can be evaluated by repeatedly subtracting 360 degrees.

Sine of 750 degrees equals sine of 30 degrees, which is 1/2.

Transcripts

Browse More Related Video

30-60-90 Triangles - Special Right Triangle Trigonometry

Cofunction Identities Examples & Practice Problems Trigonometry

Coterminal Angles In Radians & Degrees - Basic Introduction, Trigonometry

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

The Unit Circle, Basic Introduction, Trigonometry

Trigonometric Functions of Any Angle - Unit Circle, Radians, Degrees, Coterminal & Reference Angles

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
TrigonometryPeriodic FunctionsSineCosineTangentCotangentCoterminal AnglesUnit CircleMath ProblemsEducational ContentReference Angles