Short Answer 
To find missing values in a right triangle, understand the trigonometric relationships SOH CAH TOA, which represent sine, cosine, and tangent. Apply these functions by plugging in known values to set up equations for the unknown lengths and verify results to ensure accuracy with the angle-side ratios.
Step 1: Understand the Basics of SOH CAH TOA
To find the missing values in a right triangle, it’s essential to grasp the basic trigonometric relationships represented by the acronym SOH CAH TOA. Here’s what they stand for:
- SOH: Sine (sin) = opposite / hypotenuse
- CAH: Cosine (cos) = adjacent / hypotenuse
- TOA: Tangent (tan) = opposite / adjacent
These functions allow us to relate the angles and sides of the triangle, helping to solve for unknown lengths.
Step 2: Apply Trigonometric Functions to Solve for Missing Values
Once you understand the relationships, apply them to find the unknown lengths by plugging in the known values. For example:
- For sin 45: Set up the equation as sin 45 = 12 / x to solve for x.
- For sin 45 again: Rearrange it to find x = 12 / sin 45.
- For tan 30: Use the formula tan 30 = 11 / x to find x.
This will provide you with the respective lengths based on each calculation.
Step 3: Verify Results and Additional Calculations
After calculating the values for each triangle side, it’s important to verify the results to ensure accuracy. For instance:
- Check if x = 12‚Äöao25 holds true by substituting back in the sine equation.
- Verify x = 3‚Äöao36 with the sine ratio.
- Also, calculate x = 18 cos 60 to ensure it matches the cosine relationship.
This final verification step will confirm that all calculated values align correctly with the triangle’s angle-side ratios.