Trig - 0.7 Trig Functions of Any Angle

Kimberly Brehm
26 Jun 202108:15
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, the concept of trigonometric functions for any angle is explored through the lens of the unit circle. The presenter begins by reviewing the basics of the unit circle and how it relates to right triangles, establishing the foundation for understanding trigonometric functions. The video then delves into the implications of considering angles in different quadrants, emphasizing the importance of recognizing the signs of the x and y coordinates in each quadrant to determine the appropriate values for sine, cosine, and tangent. The presenter uses the mnemonic 'All Students Take Calculus' to aid in remembering the signs of these functions across quadrants. Practical examples are provided, illustrating how to calculate the sine and cotangent of an angle given a point's coordinates in the fourth quadrant. The video concludes with an invitation for viewers to practice similar problems and hints at future content based on student needs.

Takeaways
  • 📐 Remember the unit circle and its relation to trigonometric functions, where x and y represent the coordinates on the circle, and θ (theta) is the central angle.
  • 🔢 The radius (r) of the unit circle is always positive, which helps determine the sign of trigonometric function values based on the quadrant.
  • 📉 For the first quadrant, all ordered pairs (x, y) are positive, making sine, cosine, and tangent positive.
  • 📈 In the second quadrant, x is negative and y is positive, which affects the signs of the trigonometric functions accordingly.
  • 📉📉 The third quadrant has both x and y as negative, again influencing the signs of the trigonometric ratios.
  • 📈 The fourth quadrant features a positive x and a negative y, which is crucial for determining the signs of sine and cosine.
  • ➗ To find the hypotenuse (r), use the Pythagorean theorem: r^2 = x^2 + y^2, ensuring you take the positive root for r.
  • 📌 Sine (sin) is defined as the y-coordinate over the radius (opposite over hypotenuse), cosine (cos) as the x-coordinate over the radius (adjacent over hypotenuse).
  • 🔁 Cotangent (cot) is the reciprocal of tangent (tan), which is the x-coordinate over the y-coordinate (adjacent over opposite).
  • 🧮 Use the acronym 'ALL Students Take Calculus' to remember the signs of the trigonometric functions in each quadrant.
  • ✅ Practice finding the trigonometric function values for points in different quadrants by first determining θ' (theta prime) and then using the right triangle formed.
  • 📚 Understanding the signs of trigonometric functions is essential for accurately calculating their values in various quadrants.
Q & A

  • What is the significance of the unit circle in the context of trigonometric functions?

    -The unit circle is a circle with a radius of one and is used to define trigonometric functions for all angles. It helps in understanding the relationship between the angles and the coordinates of a point on the circle, which are used to define sine, cosine, and tangent.

  • How does the position of a point in a quadrant affect the signs of its coordinates?

    -In the first quadrant, both x and y coordinates are positive. In the second quadrant, x is negative and y is positive. In the third quadrant, both x and y are negative. In the fourth quadrant, x is positive and y is negative.

  • What is the formula for finding the hypotenuse (r) in a right triangle when given the coordinates (x, y)?

    -The formula to find the hypotenuse (r) is r = √(x² + y²), where x and y are the lengths of the other two sides of the right triangle.

  • How does the reference angle help in finding the trigonometric functions for any angle?

    -The reference angle (θ') is the acute angle formed by the terminal side of the given angle and the x-axis. It allows us to use the properties of right triangles to find the sine, cosine, and tangent of the original angle by considering the corresponding acute angle.

  • What is the acronym 'ALL STudents Take Calculus' used for in the context of trigonometric functions?

    -The acronym 'ALL STudents Take Calculus' is a mnemonic to help remember the signs of the trigonometric functions in different quadrants. 'A' stands for All (all trigonometric functions are positive in the first quadrant), 'L' for Left (sine is positive, others are negative in the second quadrant), 'S' for So (tangent is positive, others are negative in the third quadrant), and 'T' for Take (cosine is positive, others are negative in the fourth quadrant).

  • How do you find the sine of an angle when given a point in the fourth quadrant?

    -To find the sine of an angle when given a point in the fourth quadrant, you first find the hypotenuse (r) using the coordinates of the point. Then, you calculate the sine as the ratio of the y-coordinate (opposite side) to the hypotenuse (r).

  • What is the relationship between the trigonometric functions and their reciprocals, such as cosecant, secant, and cotangent?

    -The reciprocal trigonometric functions are derived from the original trigonometric functions. Cosecant is the reciprocal of sine (cosecant = 1/sine), secant is the reciprocal of cosine (secant = 1/cosine), and cotangent is the reciprocal of tangent (cotangent = 1/tangent).

  • How does the sign of the y-coordinate affect the sign of the sine function for a given angle?

    -The sign of the sine function is determined by the sign of the y-coordinate. If the y-coordinate is positive, the sine is positive; if the y-coordinate is negative, the sine is negative. This is because sine represents the ratio of the y-coordinate (opposite side) to the hypotenuse (r) in a right triangle.

  • What is the process for finding the cotangent of an angle when given a point in a specific quadrant?

    -To find the cotangent of an angle, you first identify the x and y coordinates of the point, which correspond to the adjacent and opposite sides of the right triangle, respectively. Then, you calculate the cotangent as the ratio of the x-coordinate (adjacent side) to the y-coordinate (opposite side).

  • How does the concept of a reference angle simplify the process of finding trigonometric function values for any angle?

    -A reference angle simplifies the process by allowing us to consider only the acute angle formed by the terminal side of the given angle and the x-axis. This means we can use the properties of right triangles and the known values of trigonometric functions for acute angles to find the values for any angle.

  • What is the significance of the hypotenuse (r) always being positive in the context of trigonometric functions?

    -The hypotenuse (r) always being positive ensures that the trigonometric function values are consistent and predictable, regardless of the quadrant in which the angle lies. It also helps in determining the correct signs of the trigonometric function values based on the quadrant.

Outlines
00:00
📐 Understanding Trigonometric Functions with Any Angle

This paragraph introduces the concept of trigonometric functions for any angle using the unit circle as a reference. It explains that for an angle theta, a perpendicular line is drawn from a point on the circle to create a right triangle. The coordinates (x, y) and the radius (r) are used to define the sine, cosine, and tangent functions. The paragraph also discusses how to handle angles in different quadrants by considering the signs of the coordinates. An example is given using a point in the fourth quadrant to illustrate the process of finding sine and cotangent of an angle using the reference angle theta prime. The importance of understanding the signs of the trigonometric functions in each quadrant is emphasized, with the acronym 'ALL STUDENTS' provided as a mnemonic to remember the signs.

05:04
📉 Applying Trigonometric Functions to Points in Different Quadrants

The second paragraph delves deeper into applying trigonometric functions to points located in various quadrants. It reiterates the signs of the trigonometric functions in each quadrant using the acronym 'ALL STUDENTS' to remember that sine, cosine, and tangent have positive values in the first quadrant, while the other functions have varying signs in the subsequent quadrants. A practice example is presented where a point in the second quadrant is used to find the sine, cosine, and tangent of the angle theta prime. The paragraph also explains how to find the reciprocal trigonometric functions such as cosecant, secant, and cotangent. The video concludes with a teaser for future content based on student needs.

Mindmap
Keywords
💡Trigonometric Functions
Trigonometric functions, often abbreviated as trig functions, are mathematical functions that relate angles to the ratios of the lengths of the sides of a right triangle. In the context of the video, these functions are used to analyze angles and their relationships within a unit circle, providing a way to understand and calculate values for various angles in different quadrants.
💡Unit Circle
A unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it allows for the definition of trigonometric functions for all angles, not just those in a right triangle. In the video, the unit circle is used to illustrate how trig functions change with different angles and quadrants.
💡Right Triangle
A right triangle is a triangle in which one of the angles is a right angle (90 degrees). It is the basis for many trigonometric concepts, as the relationships between the angles and sides of a right triangle can be used to define trig functions. In the video, the properties of right triangles are used to explain how trig functions like sine, cosine, and tangent are derived.
💡Quadrants
In a two-dimensional coordinate system, the quadrants are the four regions defined by the intersection of the positive and negative x and y axes. Each quadrant is characterized by the signs of the x and y coordinates of the points within it. The video uses the concept of quadrants to explain how the signs of trig function values change depending on the location of the angle within a particular quadrant.
💡Reference Angle
A reference angle is the acute angle formed by a line and the x-axis in a coordinate system. It is used to find the values of trig functions for any angle by considering the coterminal angle that lies between 0 and 90 degrees. In the video, the reference angle is used to simplify the process of finding trig function values for angles in different quadrants.
💡Sine (sin)
The sine function, often abbreviated as sin, is a trigonometric function that gives the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle or the y-coordinate of a point on the unit circle. It is used to determine the vertical position of a point in relation to the x-axis. In the video, sine is used to find the y-value based on the reference angle.
💡Cosine (cos)
The cosine function, often abbreviated as cos, is a trigonometric function that gives the ratio of the length of the side adjacent to an angle to the length of the hypotenuse in a right triangle or the x-coordinate of a point on the unit circle. It is used to determine the horizontal position of a point in relation to the y-axis. In the video, cosine is mentioned in the context of finding the x-value based on the reference angle.
💡Tangent (tan)
The tangent function, often abbreviated as tan, is a trigonometric function that gives the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle in a right triangle or the ratio of the y-coordinate to the x-coordinate of a point on the unit circle. It is used to find the steepness or slope of a line. In the video, tangent is used to find the ratio of x to y for an angle in a right triangle.
💡Cosecant (csc)
Cosecant, often abbreviated as csc, is the reciprocal of the sine function. It gives the ratio of the hypotenuse to the length of the side opposite an angle in a right triangle or the ratio of the radius to the y-coordinate on the unit circle. It is used to find the length of the hypotenuse given the length of the opposite side. In the video, cosecant is mentioned as one of the trig functions related to sine.
💡Secant (sec)
Secant, often abbreviated as sec, is the reciprocal of the cosine function. It gives the ratio of the hypotenuse to the length of the side adjacent to an angle in a right triangle or the ratio of the radius to the x-coordinate on the unit circle. It is used to find the length of the hypotenuse given the length of the adjacent side. In the video, secant is mentioned as one of the trig functions related to cosine.
💡Cotangent (cot)
Cotangent, often abbreviated as cot, is the reciprocal of the tangent function. It gives the ratio of the length of the side adjacent to an angle to the length of the side opposite an angle in a right triangle or the ratio of the x-coordinate to the y-coordinate on the unit circle. It is used to find the ratio of the adjacent side to the opposite side given the slope of a line. In the video, cotangent is used to find the ratio of x to y for an angle in a right triangle.
Highlights

Review of trigonometric functions for any angle, using the unit circle and right triangles.

Explanation of sine and cosine in terms of opposite over hypotenuse and adjacent over hypotenuse, respectively.

Introduction of a new angle, theta prime, to handle angles outside the standard position.

Use of the right triangle to find sine and cotangent of theta prime.

Calculation of the hypotenuse (r) for a given point in a quadrant.

Determination of the sign of trigonometric functions based on the quadrant.

Application of the TAKE CALCULUS acronym to remember the signs of trigonometric functions in each quadrant.

Example calculation using a point in the fourth quadrant (20, -21) to find sine and cotangent.

Process of finding the hypotenuse (r) when given the x and y coordinates of a point.

Explanation of how to find the sine, cosine, and tangent of an angle using the unit circle.

Introduction of reciprocal trigonometric functions: cosecant, secant, and cotangent.

Practice example given for a point in a different quadrant (-3, 4) to reinforce understanding.

Emphasis on the importance of drawing the triangle for visual and conceptual understanding.

Mention of future videos for a review of trigonometric functions based on student needs.

Use of the unit circle to understand the behavior of trigonometric functions for various angles.

Understanding the concept of reference angles and how they simplify the process of finding trigonometric values.

The significance of the unit circle's radius (r) always being positive when determining function signs.

Practical application of the quadrant system to determine the signs of trigonometric function values.

The use of the TAKE CALCULUS acronym as a mnemonic device for quadrant sign patterns.

Transcripts

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