The Signs of Trigonometric Functions(All Students Take Calculus)

The Math Sorcerer

15 May 201804:18

EducationalLearning

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TLDRThe video script discusses the signs of trigonometric functions across different quadrants. It explains that in the first quadrant, all trig functions are positive, while in the second and third quadrants, only certain functions maintain their positivity. The mnemonic 'all students take calculus' is used to remember the signs, with each word representing a quadrant and the initial letter corresponding to the sign of the trig functions. The video also provides a method to solve problems by identifying the quadrant based on the signs of sine and secant.

Takeaways

📈 In the first quadrant, all trigonometric functions (sine, cosine, and tangent) are positive.
📌 Sine (sin(θ)) represents the y-coordinate over the hypotenuse (R), cosine (cos(θ)) is the x-coordinate over R, and tangent (tan(θ)) is y/x.
🔄 The signs of the trig functions change with each quadrant: positive in Q1, negative x in Q2, negative y in Q3, and positive x and y in Q4.
💡 The mnemonic 'all students take calculus' is used to remember the signs of the trig functions in each quadrant.
📊 In the second quadrant, cosine is negative and sine is positive, leading to a negative tangent.
📐 In the third quadrant, both sine and cosine are negative, but tangent is positive because it's the division of two negatives.
🌟 In the fourth quadrant, only cosine is positive, while sine and tangent are negative, reinforcing the mnemonic's reference to 'calculus'.
🤓 The mnemonic 'all students take calculus' can be visualized as 'all' for sine, 'students' for tangent, and 'calculus' for cosine, which correspond to the first letters of 'A', 'S', and 'C'.
🔑 To find the quadrant for a given trig function value, use the mnemonic and the signs of the functions to deduce the correct quadrant.
📚 The video provides an example of using the mnemonic to determine the quadrant where sine is negative and secant (the reciprocal of cosine) is positive, which is the fourth quadrant.

Q & A

What are the main topics discussed in the video?
-The main topics discussed in the video are the signs of trigonometric functions and how to remember them across different quadrants of the Cartesian coordinate system.
How are trigonometric functions positive in the first quadrant?
-In the first quadrant, all trigonometric functions are positive because both the x and y coordinates (R) are positive, resulting in positive values for sine (θ), cosine (θ), and tangent (θ).
What is the relationship between sine, cosine, and the coordinates in the first quadrant?
-In the first quadrant, sine (θ) is represented as y/R, cosine (θ) as x/R, and tangent (θ) as y/x. Since both x and y are positive, all these functions are positive.
How does the sign of trigonometric functions change in the second quadrant?
-In the second quadrant, cosine (θ) is negative because x is negative, while sine (θ) remains positive as y is positive. Tangent (θ) becomes negative because it involves dividing a positive by a negative.
What are the mnemonics provided in the video to remember the signs of trigonometric functions?
-The video provides a mnemonic: 'all students take calculus' to remember that in the first quadrant, all trigonometric functions (sine, cosine, tangent) are positive.
How does the sign of cosine differ between the first and second quadrants?
-Cosine is positive in the first quadrant but becomes negative in the second quadrant due to the negative x-coordinate.
What is the significance of the quadrant where sine is negative and secant is positive?
-The quadrant where sine is negative and secant (the reciprocal of cosine) is positive is the fourth quadrant, as it implies that cosine is positive and sine is negative.
How does the video use humor to help memorize the signs of trigonometric functions?
-The video uses a play on words with 'all students take calculus' as a humorous way to remember that in the first quadrant, all trigonometric functions are positive.
What is the relationship between the signs of the trigonometric functions and the signs of their reciprocals (cosecant, secant, and cotangent)?
-The signs of the reciprocals (cosecant, secant, and cotangent) are the same as the signs of the original trigonometric functions (sine, cosine, tangent) in the first quadrant.
How does the sign of tangent change in the third quadrant?
-In the third quadrant, tangent (θ) is positive because it involves dividing a negative y by a negative x, which results in a positive value.
What is the method to find the quadrant where sine of theta is less than zero and secant of theta is greater than zero?
-To find the quadrant where sine of theta is less than zero and secant of theta is greater than zero, you look for the quadrant where cosine (and thus secant, as secant is the reciprocal of cosine) is positive and sine is negative, which is the fourth quadrant.

Outlines

00:00

📚 Introduction to Trigonometric Functions' Signs

This paragraph introduces the concept of signs of trigonometric functions, specifically in the context of the four quadrants of the Cartesian coordinate system. It explains that in the first quadrant, all trigonometric functions (sine, cosine, and tangent) are positive because both x and y coordinates are positive. The explanation includes a brief review of the definitions of sine, cosine, and tangent in terms of the coordinates (x, y) and the radius R from the origin. The paragraph also introduces a mnemonic device, 'all students take calculus,' to help remember the signs of the trigonometric functions and their reciprocals (cosecant, secant, and cotangent) across the different quadrants.

Mindmap

Keywords

💡Trigonometric Functions

Trigonometric functions, often abbreviated as trig functions, are mathematical functions that relate angles of a right triangle to the ratios of the lengths of its sides. In the context of the video, these functions include sine, cosine, and tangent, which are fundamental in solving various mathematical problems involving triangles and periodic phenomena.

💡Signs of Trig Functions

The signs of trig functions refer to the positive or negative values these functions take depending on the quadrant in which the angle lies. This is crucial for determining the behavior of trigonometric functions in different regions of the coordinate plane and is a key aspect of the video's content.

💡Quadrants

The quadrants are the four equal regions of the Cartesian coordinate system, each defined by the signs of the x (horizontal) and y (vertical) coordinates. Quadrants are numbered counterclockwise, starting from the upper right where both coordinates are positive. The video uses quadrants to explain the signs of trig functions.

💡Reciprocals of Trig Functions

Reciprocals of trig functions refer to the inverse operations of the basic trigonometric functions, which are cosecant (reciprocal of sine), secant (reciprocal of cosine), and cotangent (reciprocal of tangent). These are used to relate the trig functions to their inverses and are important for understanding the full range of trigonometric behavior.

💡Positive and Negative Values

Positive and negative values are fundamental concepts in mathematics that represent quantities with opposite directions or effects. In the context of the video, they are used to describe the signs of trig functions in different quadrants, which is essential for solving trigonometric equations and understanding their graphs.

💡Right Triangle

A right triangle is a triangle in which one of the angles is a right angle (90 degrees). Trigonometric functions are originally defined for angles in a right triangle, with the sides of the triangle being used to calculate the ratios that define the functions.

💡Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional coordinate system that consists of a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at a point called the origin. It is used to represent and graph functions, including trigonometric functions.

💡Sine (sin)

Sine is one of the primary trigonometric functions, defined as the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle. It is used in various mathematical and physical applications, including the representation of periodic phenomena.

💡Cosine (cos)

Cosine is another fundamental trigonometric function, defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. It plays a key role in calculating angles and is used in many areas of mathematics and science.

💡Tangent (tan)

Tangent is a trigonometric function that represents the ratio of the sine to the cosine of an angle. It is used to find the angle of a right triangle and is particularly relevant in the study of slopes and rates of change.

💡Mnemonic

A mnemonic is a device or technique that helps with memory retention, often involving patterns or phrases that are easy to remember. In the context of the video, a mnemonic is used to help recall the signs of trig functions in different quadrants.

Highlights

Introduction to the signs of trigonometric functions

All trig functions are positive in the first quadrant (Quadrant 1)

In Quadrant 1, sine (θ) is Y/R, cosine (θ) is X/R, and tangent (θ) is Y/X

Explanation of why all trig functions are positive in Quadrant 1

Reciprocals of trig functions: cosecant, secant, and cotangent

Signs of trig functions in Quadrant 2: cosine is negative, sine is positive

Signs of trig functions in Quadrant 3: sine is negative, cosine is negative, tangent is positive

Signs of trig functions in Quadrant 4: cosine is positive, sine is negative, tangent is negative

Mnemonic for remembering signs: 'All students take calculus'

Example problem: finding the quadrant where sine is less than zero and secant is greater than zero

Solution to the example problem: Quadrant 4

Explanation of how the mnemonic helps in solving the problem

The importance of understanding the signs of trig functions in solving problems

The video provides a method to remember the signs of trig functions across different quadrants

The use of a simple mnemonic to aid in memorization and quick recall of trig function signs

The video's approach to teaching trigonometry is tailored to students taking calculus

Transcripts

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Takeaways

Q & A

What are the main topics discussed in the video?

How are trigonometric functions positive in the first quadrant?

What is the relationship between sine, cosine, and the coordinates in the first quadrant?

How does the sign of trigonometric functions change in the second quadrant?

What are the mnemonics provided in the video to remember the signs of trigonometric functions?

How does the sign of cosine differ between the first and second quadrants?

What is the significance of the quadrant where sine is negative and secant is positive?

How does the video use humor to help memorize the signs of trigonometric functions?

What is the relationship between the signs of the trigonometric functions and the signs of their reciprocals (cosecant, secant, and cotangent)?

How does the sign of tangent change in the third quadrant?

What is the method to find the quadrant where sine of theta is less than zero and secant of theta is greater than zero?