Trigonometry

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THERE ARE TWO special triangles in trigonometry. One is the 30°-60°-90° triangle. The other is the isosceles right triangle. They are special because with simple geometry we can know the ratios of their sides, and therefore solve any such triangle.

Theorem.  In a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : square root of 3.

30-60-90 triangle

We will prove that below.

Note that the smallest side, 1, is opposite the smallest angle, 30°; while the largest side, 2, is opposite the largest angle, 90°.  (Theorem 6). (For, 2 is larger than square root of 3.  Also, while 1 : square root of 3: 2 correctly corresponds to the sides opposite 30°-60°-90°, many find the sequence 1 : 2 : square root of 3 easier to remember.)

The cited theorems are from the Appendix, Some theorems of plane geometry.

Here are examples of how we take advantage of knowing those ratios.  First, we can evaluate the functions of 60° and 30°.

Example 1.   Evaluate cos 60°.

Answer.  For any problem involving a 30°-60°-90° triangle, the student should not use a table.  The student should sketch the triangle and place the ratio numbers.

30-60-90 triangle

Since the cosine is the ratio of the adjacent side to the hypotenuse, we can see that

cos 60° = ½.

Example 2.   Evaluate sin 30°.

Answer.   According to the property of cofunctions, sin 30° is equal to cos 60°.   sin 30° = ½.

On the other hand, you can see that directly in the figure above.

Problem 1.   Evaluate sin 60° and tan 60°.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

The sine is the ratio of the opposite side to the hypotenuse.

    sin 60° =  square root of 3
 2
 = ½square root of 3.

Lesson 5 of Algebra

The tangent is ratio of the opposite side to the adjacent.

    tan 60° =  square root of 3
 1
= square root of 3.

Problem 2.   Evaluate cot 30° and cos 30°.

1 : 2: SqRt 3

The cotangent is the ratio of the adjacent side to the opposite.

   Therefore, on inspecting the figure above, cot 30° =  square root of 3
 1

= square root of 3.

Or, more simply, cot 30° = tan 60°.

Problem 1

As for the cosine, it is the ratio of the adjacent side to the hypotenuse. Therefore,

cos 30° =  square root of 3
 2
 = ½square root of 3.

Before we come to the next Example, here is how we relate the sides and angles of a triangle:

Triangle ABC

If an angle is labeled capital A, then the side opposite will be labeled small a.  Similarly for angle B and side b, angle C and side c.

Example 3.   Solve the right triangle ABC if angle A is 60°, and side AB is 10 cm.

1:2:SqRt3

Solution.  To solve a triangle means to know all three sides and all three angles.  Since this is a right triangle and angle A is 60°, then the remaining angle B is its complement, 30°.

Again, in every 30°-60°-90° triangle, the sides are in the ratio 1 : 2 : square root of 3, as shown on the left.

When we know the ratios of the sides, then to solve a triangle we do not require the trigonometric functions or the Pythagorean theorem. We can solve it by the method of similar figures.

Now, the sides that make the equal angles are in the same ratio. Proportionally,

2 : 1 = 10 : AC.

2 is two times 1. Therefore 10 is two times AC.  AC is 5 cm.

The side adjacent to 60°, we see, is always half the hypotenuse.

As for BC—proportionally,

2 : square root of 3 = 10 : BC.

To produce 10,  2 has been multiplied by 5. Therefore, square root of 3 will also be multiplied by 5.  BC is 5square root of 3 cm.

In other words, since one side of the standard triangle has been multiplied by 5, then every side will be multiplied by 5.

1:2:SqRt3

1 : 2 : square root of 3 = 5 : 10 : 5square root of 3.

Compare Example 11 here.

Again: When we know the ratio numbers, then to solve the triangle the student should use this method of similar figures, not the trigonometric functions.

(In Topic 10, we will solve right triangles whose ratios of sides we do not know.)

Problem 3.   In the right triangle DFE, angle D is 30° and side DF is 3 inches.  How long are sides d and f ?

1 : 2: Sq rt 3

1 : 2: SqRt 3

   The student should draw a similar triangle in the same orientation. Then see that the side corresponding to square root of 3 was multiplied by square root of 3.

Lesson 26 of Algebra

Therefore, each side will be multiplied by square root of 3. Side d will be
1timessquare root of 3 = square root of 3. Side f will be 2square root of 3.

Problem 4.   In the right triangle PQR, angle P is 30°, and side r is 1 cm.  How long are sides p and q ?

1 : 2: SqRt 3

1 : 2: SqRt 3

The side corresponding to 2 has been divided by 2.   Therefore, each side must be divided by 2.  Side p will be ½, and side q will be ½square root of 3.

Problem 5.   Solve the right triangle ABC if angle A is 60°, and the hypotenuse is 18.6 cm.

30-60-90 triangle

30-60-90 triangle

The side adjacent to 60° is always half of the hypotenuse -- therefore, side b is 9.3 cm.
But this is the side that corresponds to 1. And it has been multiplied by 9.3.  Therefore, side a will be multiplied by 9.3.
It will be 9.3square root of 3 cm.

Problem 6.    Prove:  The area A of an equilateral triangle whose side is s, is

A = ¼square root of 3s2.

30-60-90 triangle

The area A of any triangle is equal to one-half the sine of any angle times the product of the two sides that make the angle. (Topic 2, Problem 6.)

In an equilateral triangle each side is s , and each angle is 60°.  Therefore,

A = ½ sin 60°s2.

Since sin 60° = ½square root of 3,

Problem 1

A = ½· ½square root of 3 s2 = ¼square root of 3s2.

Problem 7.    Prove:  The area A of an equilateral triangle inscribed in a circle of radius r, is

A 3
4
square root of 3 r2.

30-60-90 triangle

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Click Custom and navigate to your specific driver file (usually an .INF file for your board like XP600 or DX5). Helpful Configuration Tips

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By following this guide, you can download and install Maintop 5.3 and start creating professional-looking labels and stickers with ease.

5. Logging and Export

All sensor data and diagnostic results can be saved to a plain-text .log or .csv file for later analysis or remote support.


Step 4: Allow Driver Installation

During installation, Windows may display a warning that the driver is unsigned (most legacy VIA drivers are). Click “Install this driver software anyway”.

2. Key Features of MainTop 5.3

  • Dual Environment: Schematic capture and PCB layout in one program.
  • Manual Routing: Full control over track placement; no auto-router.
  • Library System: Discrete components, DIP ICs, connectors, and basic SMD pads.
  • Gerber Output: Produces RS-274X Gerber files for manufacturing.
  • Lightweight: Runs on Windows 95/98/ME/2000/XP, and even on modern Windows via compatibility mode.
  • File Extensions: .pcb (board file), .sch (schematic), .lib (library).

Step 2: Installation Process

  1. Prepare the System:

    • Log in as Administrator.
    • Disable your antivirus temporarily (RIP software often modifies system ports and may be flagged incorrectly as suspicious).
  2. Run the Installer:

    • Locate the setup.exe file.
    • Right-click and select "Run as Administrator".
  3. Select Language:

    • MainTop usually prompts for a language (English or Chinese) at the start. Select English.
  4. Setup Wizard:

    • Click Next through the welcome screen.
    • Accept the License Agreement.
    • Destination Folder: It is highly recommended to install to the default path C:\MainTop or C:\DTP. Do not use spaces in the folder name.
  5. Select Components:

    • You will see options for MainTop Print, MainTop Edit, and MainTop Cut.
    • Select all that apply to your workflow. "Print" is the core RIP module.
  6. Security Key (Dongle):

    • The installer may ask if you have a dongle. If you have the USB key, insert it when prompted or wait until after installation to insert it.
    • Note: If you are using a "Soft Key" (software license), you will need to run a specific registration tool after installation.
  7. Finish:

    • Complete the installation and reboot your computer. Do not launch the software yet.

Frequently Asked Questions (FAQ)

Q1: Is there a 64-bit version of MainTop 5.3?
No. Version 5.3 was released before widespread 64-bit adoption. However, it runs perfectly on 64-bit versions of Windows 7/8/10 via WOW64 emulation.

Q2: Can I use MainTop 5.3 on Linux via Wine?
Partially. The UI works, but hardware sensors and CMOS access will fail because Wine doesn’t emulate low-level I/O ports. Use lm-sensors instead.

Q3: What’s the difference between MainTop 5.3 and 6.0?
Version 6.0 added support for VIA Nano processors and improved Windows 8 compatibility, but removed some legacy diagnostic features. Stick with 5.3 for older Eden/C3-based boards.

Q4: My antivirus deleted MainTop. What now?
Restore from quarantine, then add an exception for the file/folder. Then upload it to VirusTotal – if only 2-3 obscure AVs flag it, it’s a false positive. If 15+ flag it, delete immediately.

Q5: Where is the official download page?
VIA’s original ftp://ftp.viatech.com/ was decommissioned in 2019. The official successor is the “VIA Legacy Download Hub,” requiring a free account.


Error 3: Program crashes on Windows 10/11

Workaround:

  • Set Compatibility Mode to Windows 7
  • Disable full-screen optimizations
  • Run in 256-color mode (rarely needed but sometimes works)

Common Errors & Fixes for MainTop 5.3

Even after a successful download and install, you may encounter issues.

Problem 8.   Prove:  The angle bisectors of an equilateral triangle meet at a point that is two thirds of the distance from the vertex of the triangle to the base.

30-60-90 triangle

Let ABC be an equilateral triangle, let AD, BF, CE be the angle bisectors of angles A, B, C respectively; then those angle bisectors meet at the point P such that AP is two thirds of AD.

First, triangles BPD, APE are congruent.

For, since the triangle is equilateral and BF, AD are the angle bisectors, then angles PBD, PAE are equal and each 30°;
and the side BD is equal to the side AE, because in an equilateral triangle the angle bisector is the perpendicular bisector of the base.

Theorem 2

Angles PDB, AEP then are right angles and equal.

Therefore,

Angle-side-angle

triangles BPD, APE are congruent.

30-60-90 triangle

  Now,  BP
PD
= csc 30° = 2.

Problem 2

Therefore, BP = 2PD.

But AP = BP, because triangles APE, BPD are conguent, and those are the sides opposite the equal angles.
Therefore, AP = 2PD.
     Therefore AP is two thirds of the whole AD.
Which is what we wanted to prove.

The proof

Here is the proof that in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : square root of 3.  It is based on the fact that a 30°-60°-90° triangle is half of an equilateral triangle.

Draw the equilateral triangle ABC.  Then each of its equal angles is 60°.  (Theorems 3 and 9)

A 30-60-90 triangle

Draw the straight line AD bisecting the angle at A into two 30° angles.
Then AD is the perpendicular bisector of BC  (Theorem 2).  Triangle ABD therefore is a 30°-60°-90° triangle.

Now, since BD is equal to DC, then BD is half of BC.

This implies that BD is also half of AB, because AB is equal to BC.  That is,

BD : AB = 1 : 2

From the Pythagorean theorem, we can find the third side AD:

AD2 + 12 = 22
AD2 = 4 − 1 = 3
AD = square root of 3.

Therefore in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : square root of 3; which is what we set out to prove.

Corollary.  The square drawn on the height of an equalateral triangle is three fourths of the square drawn on the side.


Next Topic:  The Isosceles Right Triangle


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