Alternative University

Math Preliminaries


Trigonometry calculates the horizontal and vertical distance of a unit circle point that is specified by an angle.  The angle is the perimeter distance to the point on the circle, from a predetermined starting point on the circle, which in our examples will be the East (3 O’Clock) position.

As explained in the previous lesson o, we will be using a right handed coordinate system with angles measured counter-clockwise (CCW) from East:

Figure 1: Right handed coordinate system measures angles CCW from East.

We divide the coordinate system into four quadrants, in the counter-clockwise order of traversing the unit circle perimeter.  The first quadrant includes North East directions (NE), the second quadrant NW, the third quadrant SW, and the fourth quadrant SE:


The first quadrant is subtended by the angle 𝛑╱𝟐 (90 degrees):

Figure 2: First quadrant is subtended by π ⁄ 2.

We assign the symbol theta ( 𝜃 ) to denote an angle that is the perimeter distance along the unit circle measured counter-clockwise (CCW) from East:

Figure 3: Theta denotes an angle (arc length).

If the angle is in the first quadrant, as shown here, it is called an acute angle o.  Trigonometry finds the horizontal and vertical position of any point on the unit circle at any angle (in any quadrant).

In this discussion, we denote the horizontal (eastward) position of a point as x, and the vertical (northward) position as y:

Figure 4: Horizontal and vertical distances x and y from unit circle origin to a point on the circle. Theta is the “angle” (arc) of that point (distance along the circle perimeter to that point).

The horizontal distance x is referred to as the cosine (“cos”) of the point at theta, and the vertical distance y is referred to as the sine (“sin”) of the point:

x  =  cos𝜃
y  =  sin𝜃

Cosine and sine are pronounced co-sign and sign respectively (not spelled that way, just pronounced that way):

Pronounciation of “sine”. [Wikimedia]

The following formulas may be used to calculate cosine and sine:

cos𝜃 = 1 − (𝜃2 ⁄ 2!) + (𝜃4 ⁄ 4!) − (𝜃6 ⁄ 6!) + ⋯
sin𝜃 = 𝜃 − (𝜃3 ⁄ 3!) + (𝜃5 ⁄ 5!) − (𝜃7 ⁄ 7!) + ⋯

where the exclamation mark (!) is the factorial symbol, which multiplies a number with all positive integers that are less than the number. For example:

5! = 5 × 4 × 3 × 2 × 1

These types of formulas are best left to calculators and computers. Modern calculators have buttons for sine and cosine.

Figure 5: Texas Instruments handheld calculator with SIN and COS buttons. [Wikimedia]

Figure 6: On-screen calculator (built-in user app) that comes with Microsoft Windows includes “sin” and “cos” buttons.

Inverse Trigonometric Functions

The inverse of the cosine and sine functions are arccosine and arcsine, which give us the angle of a given horizontal and vertical distance respectively.

“Think of arcsin as meaning ‘the arc whose sine is’ or ‘the angle whose sine is’.”
Varberg, Purcell & Rigdon, Calculus 9th ed.

The inverse function arccos may also be denoted as cos−1, and arcsin may be denoted sin−1

arccos    cos−1

arcsin    sin−1

Formulas are available for calculating inverse trigonometric functions.

  arcsin( y ) = 
y + ( 1 ⁄ 2 ) ( y3 ⁄ 3 )
   + ( ( 1 · 3 ) ⁄ ( 2 · 4 ) ) · ( y5 ⁄ 5 )
   + ( ( 1 · 3 · 5 ) ⁄ ( 2 · 4 · 6 ) ) · ( y7 ⁄ 7 )
   + ···

But, again, such tedious numerics are best left to calculators and computers. We usually only need to know the sine or cosine, or arc length (angle) of a sine or cosine, not how it’s calculated.

On calculators, often the same buttons are used for forward and inverse trigonometric functions, except that a shift key is pressed (and released) beforehand to notify the calculator to use inverse instead of forward trigonometry.

For example, on the Texas Instruments handheld calculator pictured, punching in a number and then pressing the SIN key calculates the sine of that number, but pressing (and releasing) the 2ND key right before pressing the SIN key invokes the arcsine function:

Figure 7:  Close-up of some of the buttons on the Texas Instruments calculator. Each button has a description on the button itself, and separate descriptions above each button for when the 2ND and/or ALPHA buttons are pressed beforehand.

On this calculator, a blue description (upper left above each button) is invoked if the 2ND key is pressed beforehand. The green description (a text character to the right of the blue description) is invoked if the ALPHA key is pressed beforehand (to enter a letter of the alphabet).

For a calculator application program on a computer, if the buttons are on-screen (not actual physical buttons), the description on a button may change:

Figure 8: Microsoft Windows on-screen calculator before the 2ND key is clicked.

Figure 9: The same on-screen calculator after the 2ND key is clicked or tapped. The 2ND key is displayed as pressed, and the button descriptions have changed.


Problem: Calculate the sine of 60 degrees using the Microsoft Windows calculator.

Solution: The Microsoft Wndows calculator varies depending on which version of the operating system you are using. This example will use a slightly different version than illustrated above (with some of the feature differences noted).

Figure 10: Microsoft Windows calculator that will be used in this example.

If the calculator does not have “sin” and “cos” buttons, click on View (or the menu icon in the other calculator) and select “Scientific”.

Then make sure the type of angles is set to Degrees, as shown here. In the other on-screen calculator illustrated earlier, if it says RAD or GRAD, click/tap on that icon until it says DEG.

Next, press the ‘6’ number key, then the ‘0’ number key on your keyboard (or click/tap on the ‘6’ and ‘0’ screen buttons on the calculator).

Then click or tap on the “sin” calculator screen button (not the “sinh” button which would be for hyperbolic functions).

Figure 11: Sine of 60 degrees is 0.866…

To find the inverse, click on “Inv” then click on the arcsine button.

Figure 12: After the “Inv” button is clicked, it is displayed as pressed, and the trigonometric function buttons have changed their descriptions to inverse trigonometric functions.

Figure 13: After the arcsine button is clicked (and the arcsine is calculated as shown here), the “Inv” button automatically displays as unpressed and the trigonometric function buttons reset their descriptions to forward trigonometric functions.

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2021–Sep–22  19:46  UTC