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Triangles in Action: Applying SOHCAHTOA to Robotic Arm Design
Triangles in Action: Applying SOHCAHTOA to Robotic Arm Design
How does selecting the correct trigonometric ratio ensure our robot arm calculations are accurate?
Learning Intentions:
Sort robot arm diagrams into sine, cosine, or tangent categories based on what is known and unknown, and justify the ratio choice before calculating.
Today's Concept:
You cannot solve a math problem you haven't defined. This is the moment where the empathy work from Lessons 2–6 becomes an engineering document.
SOHCAHTOA gives you three ratios. Picking the wrong one gives you a wrong answer even when your algebra is perfect. The math looks fine; the answer is garbage.
The decision rule is simple: ask what you have and what you need. Opposite and hypotenuse? Use sine. Adjacent and hypotenuse? Use cosine. Opposite and adjacent? Use tangent.
Imagine a simple robotic arm with two links. When the arm extends, it forms a triangle. If one of the links is positioned horizontally and the other vertically, a right triangle is formed.
ABB's 7720 Articulated Robot
The two links of the arm form the legs of the triangle, while the imaginary line connecting the starting point to the final position of the arm is the hypotenuse.
Let's label the parts of the triangle formed by the arm's movement:
The base of the arm (the vertical movement) is the adjacent side.
The lift of the second link (the horizontal movement) is the opposite side.
The arm segment itself (the longest side) is the hypotenuse.
The angle at the base is θ.
SOHCAHTOA: What It Means and When to Use It
SOHCAHTOA: What It Means and When to Use It
The acronym SOHCAHTOA is a mnemonic device to help you remember the definitions of sine, cosine, and tangent.
SOH (Sine=Opposite/Hypotenuse): Use the sine function when you know or want to find the opposite side and the hypotenuse.
CAH (Cosine=Adjacent/Hypotenuse): Use the cosine function when you know or want to find the adjacent side and the hypotenuse.
TOA (Tangent=Opposite/Adjacent): Use the tangent function when you know or want to find the opposite side and the adjacent side.
You can use these functions to solve for an unknown side length or an unknown angle.
To find a missing angle, you'll need to use the inverse trigonometric functions arcsin, arccos, arctan (sin -1, cos-1 , tan-1).
How to Find an Angle Using Trigonometry
How to Find an Angle Using Trigonometry
Finding an angle in a right triangle when you know two of its side lengths is a straightforward process using inverse trigonometric functions. Here's a step-by-step guide.
Step 1: Identify the Known Sides
Step 1: Identify the Known Sides
Identify which two sides you know in relation to the angle you want to find (θ).
Hypotenuse: The side opposite the 90° angle. It's always the longest side.
Opposite: The side directly across from the angle θ.
Adjacent: The side next to the angle θ that is not the hypotenuse.
Step 2: Choose the Correct Trig Function
Step 2: Choose the Correct Trig Function
The acronym SOHCAHTOA helps you decide which function to use based on the sides you've identified.
If you know the Opposite and Hypotenuse, use Sine (SOH).
If you know the Adjacent and Hypotenuse, use Cosine (CAH).
If you know the Opposite and Adjacent, use Tangent (TOA).
Step 3: Set up the Equation
Step 3: Set up the Equation
Once you've chosen the correct function, write out the equation. For example, if you know the opposite and adjacent sides, you would write:
Step 4: Use the Inverse Function to Solve for the Angle
Step 4: Use the Inverse Function to Solve for the Angle
To isolate the angle (θ), you need to use the inverse of the trig function (arcsin, arccos, arctan). Inverse functions are denoted with a "-1" superscript (e.g., sin -1, cos-1 , tan-1).
You'll find these inverse functions on your calculator, usually by pressing a "2nd" or "Shift" key followed by the sine, cosine, or tangent button.
Step 5: Calculate the Angle
Step 5: Calculate the Angle
Plug the values into your calculator to get the final angle.
Example: Imagine you have a robotic arm with an opposite side (vertical lift) of 4 cm and an adjacent side (horizontal distance) of 3 cm. You want to find the angle (θ) of the arm's joint.
Known sides: Opposite (4 cm) and Adjacent (3 cm).
Trig function: Use Tangent (TOA) because you have the opposite and adjacent sides.
Equation: tan(θ)=4/3
Inverse function: θ=tan-1(4/3)
Calculate: θ≈53.13°
So, the angle of the robotic arm's joint would be approximately 53.13°
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