# How To Evaluate Trigonometric Functions Using Periodic Properties - Trigonometry

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

##### 📚 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

##### 💡Periodic properties

##### 💡Coterminal angles

##### 💡Unit circle

##### 💡Reference angle

##### 💡Sine function

##### 💡Cosine function

##### 💡Tangent function

##### 💡Cotangent function

##### 💡Evaluating trigonometric functions

###### 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

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