Text description - screencast for exercise 3

[The narrator reads out the onscreen text.]

NARRATOR: Exercise 3. Find the exact value of: a, sin 210 degrees, b, cos of 315 degrees, and c, tan of 150 degrees.

[A circle is drawn on an x and y graph, its centre is the intersection of the axes, which divide the circle into quadrants. A blue line runs diagonally upwards from left to right, bisecting the circle. A red curved arrow marked 210 degrees runs up from the x-axis, across the blue line and the y-axis, then down across the x-axis until it meets the blue line again. The lower point where the blue line crosses the circle is marked (cos 210 degrees, sin 210 degrees).]

NARRATOR: Part a. Find the exact value of sin of 210 degrees. So we first establish where in the unit circle 210 degrees lies, and it lies in the third quadrant. So the coordinates of this point where this line of 210 degrees meets the unit circle, this point has coordinates (cos 210 degrees, sin 210 degrees). And we note that this line is 30 degrees away from the x-axis so we should be able to relate these sin and cos of 210 degree values to sin and cos of 30 degrees.

[On both sides of the y-axis, the blue line forms an angle of 30 degrees with the x-axis. The upper point where the blue line crosses the circle is marked (cos 30 degrees, sin 30 degrees).]

NARRATOR: So this point in the first quadrant has an x-coordinate of cos 30 degrees and a y-coordinate of sin 30 degrees.

[Horizontal dotted lines run from the two points to the y-axis. The upper y-axis point is marked sin 30 degrees, the lower y-axis point is marked sin 210 degrees.]

NARRATOR: So relating these two y-coordinates, we can see from the symmetry here that sin of 210 degrees is equal to negative sin of 30 degrees. We've established from the unit circle that sin 210 degrees is equal to negative sin 30 degrees. And sin 30 degrees is an exact value that can be calculated from the following "special triangle", that is a "special triangle" that is an equilateral triangle of side length of 2.

[In a diagram, an equilateral triangle is bisected so that one side forms a right angled triangle. The vertical side is marked square root of 3, the horizontal side is marked 1 and the hypotenuse is marked 2. The upper angle is marked 30 degrees and the lower angle is marked 60 degrees.]

NARRATOR: Using trigonometry in this right angled "special triangle", we can see that sin of 30 degrees is equal to opposite of a hypotenuse which is 1 over 2. So sin of 210 degrees is equal to negative sin of 30 degrees which is equal to negative one-half.

[A circle is drawn on an x and y graph, its centre is the intersection of the axes, which divide the circle into quadrants. A blue line runs diagonally upwards from left to right, bisecting the circle. A red curved arrow marked 315 degrees runs up from the x-axis, across the blue line and the y-axis, then down across the x-axis until it meets the blue line again. The lower point where the blue line crosses the circle is marked (cos 315 degrees, sin 315 degrees).]

NARRATOR: Part b. Find the exact value of cos of 315 degrees. So we first establish where in the unit circle an angle of 315 degrees lies, and we see that it's in the third quadrant. So this point on the unit circle has an x-coordinate of cos of 315 degrees and a y-coordinate of sin of 315 degrees. We note that this line is 45 degrees away from the x-axis so we should be able to relate the values of sin and cos of 315 degrees to sin and cos of 45 degrees.

[On both sides of the y-axis, the blue line forms an angle of 45 degrees with the x-axis. The upper point where the blue line crosses the circle is marked (cos 45 degrees, sin 45 degrees). Vertical dotted lines run from the two points to the x-axis. The upper left-hand x-axis point is marked cos 315 degrees, the right-hand x-axis point is marked cos 45 degrees.]

NARRATOR: This point in the first quadrant has coordinates (cos 45 degrees, sin 45 degrees). And so we can see from symmetry that cos of 315 degrees must be equal to negative cos of 45 degrees. So from the unit circle, we've established that cos of 315 degrees equals negative cos of 45 degrees. And cos of 45 degrees is an exact value that can be calculated from the following "special triangle". And this is a right angled isosceles triangle with equal sides of length 1.

[In a diagram, a right angled triangle has sides marked 1 and 1 and a hypotenuse of square root of 2. Both angles are marked 45 degrees.]

NARRATOR: Using trigonometry in this right angled "special triangle", we can see that cos of 45 degrees is equal to adjacent over hypotenuse, which is one over square root of 2. So cos of 315 degrees equals negative cos of 45 degrees which equals negative 1 over root 2.

[A circle is drawn on an x and y graph, its centre is the intersection of the axes, which divide the circle into quadrants. A blue line runs diagonally downwards from right to left, bisecting the circle. A red curved arrow marked 150 degrees runs up from the x-axis, across the y-axis, then down until it meets the blue line. A vertical dotted black line runs down the right-hand edge of the circle, crosses the x-axis, then the blue line.]

NARRATOR: Part c. Find the exact value of tan of 150 degrees. So we first establish where in the unit circle an angle of 150 degrees lies, and we see it lies in the second quadrant. Now, tan is a little different than sin and cos. And tan is the y-coordinate of the point where this line meets with the tangent at the point (1, 0).

[The point where the dotted tangent crosses the blue line is marked (1, tan 150 degrees).]

NARRATOR: So this point down here has an x-coordinate of 1 and a y-coordinate of tan 150 degrees.

[To the left of the y-axis, the angle between the blue line and the x-axis is marked 30 degrees. To the right, a dotted line diagonal line forms a 30 degree angle with the x-axis. The point where this line crosses the tangent is marked (1, tan 30 degrees). Horizontal dotted lines run from the two marked points on the tangent to the y-axis. The top point on the y-axis is marked tan 30 degrees, the bottom point is marked tan 150 degrees.]

NARRATOR: Now, we note that this line is 30 degrees away from the x axis so we should be able to relate tan of 150 degrees to tan of 30 degrees. And we know that this point where the 30 degree line meets with the tangent has coordinates (1, tan of 30). So again using symmetry, we can see that tan of 150 degrees is equal to negative tan of 30 degrees. So from the unit circle, we've established this relationship. And tan of 30 degrees is an exact value that can be calculated from the following "special triangle" seen previously.

NARRATOR: And using trigonometry in this right angled "special triangle", we can see that tan of 30 degrees is opposite over adjacent, which is equal to 1 divided by square root of 3. So tan of 150 degrees equals negative tan of 30 degrees which equals negative 1 over root 3.