explain how you would solve a triangle given two side and the included angle.
law of cosines, then law of sines
Well, solving a triangle with two sides and an included angle is like solving a mystery with some clues. It's definitely a triangle who wants to keep its secrets hidden, but fear not, because I, your trusty Clown Bot, am here to shed some light on the situation!
First, we'll call the two given sides "a" and "b," and the included angle "C." To solve the mystery, we need to find the missing pieces. Let's play detective!
1. We can start by using the Law of Cosines to find the unknown third side, "c." It goes like this: c² = a² + b² - 2ab * cos(C). Oh, cosine, you sneaky character!
2. Next, we can use the Law of Sines to find one of the missing angles. It says that sin(A)/a = sin(B)/b = sin(C)/c. Don't worry, they won't go on "sine-ing" loudly—just calculate one of the unknown angles using this formula.
3. Now, detective skills are crucial! Since we have two sides and an included angle, try using the Law of Cosines again to find the other missing angle. This time, it'll look like: cos(A) = (b² + c² - a²) / (2bc). Apply the same logic to find angle B.
4. Finally, to find the remaining angle, simply subtract angle A and angle B from 180°. Remember, angles just can't hide from mathematics!
And voilà! With these cunning deductions, you have solved the triangle mystery. Congratulations, detective!
To solve a triangle given two sides and the included angle, you can use the Law of Cosines and Law of Sines. Here are the step-by-step instructions:
Step 1: Identify the given information in the triangle.
Let's say we have a triangle with sides a, b, and c, where sides a and b are known, and the included angle is given as angle C.
Step 2: Apply the Law of Cosines to find the third side, side c.
The Law of Cosines states that: c² = a² + b² - 2ab * cos(C)
Substitute the values of sides a, b, and angle C into the equation and solve for c.
Step 3: Apply the Law of Sines to find one of the unknown angles, angle A or angle B.
The Law of Sines states that: sin(A) / a = sin(B) / b = sin(C) / c
Substitute the known values of side a, b, angle C, and the value of side c found in the previous step.
Solve the equation for one of the unknown angles.
Step 4: Find the remaining angle.
To find the other unknown angle, subtract the sum of the two known angles from 180 degrees.
Angle A + Angle B + Angle C = 180 degrees.
Step 5: Check for any additional information.
If you have any other information about the triangle, such as the height or area, you can use relevant formulas to find them. Otherwise, you have successfully solved the triangle given two sides and the included angle.
Remember to use the appropriate trigonometric functions based on the known information, and be careful with units of measurement (degrees or radians) while using the formulas.
To solve a triangle with two sides and the included angle, you can use the Law of Cosines and the Law of Sines.
Here are the steps to follow:
1. Label the given sides and angle: Let's say you have side A, side B, and angle C.
2. Use the Law of Cosines to find the length of the third side. The formula is:
A² = B² + C² - 2BC * cos(A)
Plug in the known values for B, C, and A to calculate the missing side length.
3. Use the Law of Sines to find one of the remaining angles. The formula is:
sin(A) / a = sin(B) / b = sin(C) / c
Since you know two sides and the included angle, you can rearrange the formula to solve for one of the remaining angles.
4. Use the angle sum of a triangle to find the last angle. The sum of all angles in a triangle is always 180 degrees. So, subtract the two known angles from 180 to find the missing angle.
Once you have all three angles and side lengths, you have successfully solved the triangle.
Note: If you are working with a right-angled triangle (where the included angle is 90 degrees), you can directly use trigonometric functions such as sine, cosine, and tangent to find the missing side and remaining angles.