Sign of Trig Functions

Duration: 9 min

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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video provides a comprehensive lesson on the signs of trigonometric functions in the four quadrants of the coordinate plane. The instructor begins by setting up a standard Cartesian coordinate system, labeling the x and y axes, and then proceeds to define the quadrants. Key angles, including 0, π/2, π, 3π/2, and 2π, are marked on the axes to establish the boundaries of the quadrants. The core of the lesson is the explanation of the ASTC rule (All Students Take Calculus), which is a mnemonic to remember the signs of the trigonometric functions in each quadrant: All (sine, cosine, tangent) are positive in the first quadrant; Sine is positive in the second; Tangent is positive in the third; and Cosine is positive in the fourth. The video also includes a brief demonstration of the unit circle to explain the definitions of sine and cosine as the y and x coordinates of a point on the circle, respectively. Finally, the video presents a table of standard trigonometric values for angles 0°, 30°, 45°, 60°, and 90°, showing the values for sine, cosine, tangent, cotangent, secant, and cosecant, including cases where the function is undefined (N.D.).

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with the title 'Sign of Trigonometric functions' written in blue on a black background. The instructor draws a standard Cartesian coordinate system, labeling the horizontal axis as 'x' and the vertical axis as 'y'. He then marks the origin as 'O' and proceeds to label the key angles on the axes: π/2 at the top of the y-axis, π on the negative x-axis, 3π/2 at the bottom of the y-axis, and 2π on the positive x-axis. The instructor explains that these angles divide the plane into four quadrants, which are numbered I, II, III, and IV in an anti-clockwise direction, starting from the positive x-axis.

  2. 2:00 – 5:00 02:00-05:00

    The instructor labels the four quadrants as 'Ist Qua.', 'IInd Qua.', 'IIIrd Qua.', and 'IVth Qua.' in their respective positions. He then introduces the concept of 'Quadrantal Angles' and lists them as 0, π/2, π, 3π/2, and 2π. The main focus shifts to the signs of the trigonometric functions. The instructor writes the mnemonic 'ASTC' in a red box. He explains that in the first quadrant, all trigonometric functions (sin, cos, tan, cot, sec, cosec) are positive. In the second quadrant, only sine (sin) is positive. In the third quadrant, only tangent (tan) is positive. In the fourth quadrant, only cosine (cos) is positive. He uses the mnemonic 'All Students Take Calculus' to help remember this, with the first letter of each word corresponding to the functions that are positive in each quadrant.

  3. 5:00 – 8:49 05:00-08:49

    The instructor continues to elaborate on the signs of the trigonometric functions in each quadrant, writing 'sin +ve', 'cos +ve', 'tan +ve', 'cot +ve', 'sec +ve', and 'cosec +ve' in the first quadrant, and then 'sin +ve', 'cos -ve', 'tan -ve', 'cot -ve', 'sec -ve', 'cosec +ve' in the second quadrant, and so on for the other quadrants. He then draws a unit circle to explain the definitions of sine and cosine. He shows a point P on the circle with coordinates (cosθ, sinθ) and explains that sinθ = y/r and cosθ = x/r, where r is the radius. The video then transitions to a table titled 'Values at angles' which lists the values of sinθ, cosθ, tanθ, cotθ, secθ, and cosecθ for angles 0°, 30°, 45°, 60°, and 90°. The table shows that for 90°, tanθ and secθ are undefined (N.D.), and for 0°, cotθ and cosecθ are undefined (N.D.).

The video systematically teaches the sign conventions of trigonometric functions across the four quadrants of the coordinate plane. It begins with the foundational setup of the coordinate system and the definition of quadrants. The core of the lesson is the ASTC mnemonic, which provides a clear and memorable rule for determining the sign of any trigonometric function based on the quadrant in which the angle lies. The lesson is reinforced with a visual aid of the unit circle, which grounds the abstract concept in a geometric definition. The video concludes with a practical table of standard trigonometric values, which is essential for solving problems and understanding the behavior of these functions at key angles.

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