91Ó°ÊÓ

We are given the following information: 1. Perimeter of the triangle (P) = 29 2. Angle (A) = 42 degrees Since it's a right triangle, the other angle (B) should be 90 - 42 = 48 degrees.

We can use the sine, cosine, and tangent ratios to find the lengths of the sides in the right triangle. Let the side adjacent to angle A be a, the side opposite to angle A be b, and the side opposite to angle B (the hypotenuse) be c. The trigonometric ratios for angle A are as follows: 1. \(\sin A = \frac{b}{c}\) 2. \(\cos A = \frac{a}{c}\) 3. \(\tan A = \frac{b}{a}\)

The perimeter (P) is the sum of the lengths of all three sides of the triangle. So, we have: P = a + b + c Since we are given the perimeter as 29, we can write the equation as: 29 = a + b + c

We can use the perimeter equation to solve for one side in terms of the other two sides: c = 29 - a - b Now, we can substitute this expression of c into the sine and cosine equations in Step 2.

Substitute the expression of c from Step 4 into the sine and cosine equations in Step 2: 1. \(\sin A = \frac{b}{29 - a - b}\) => \(b = (29 - a - b) \sin 42^\circ\) 2. \(\cos A = \frac{a}{29 - a - b}\) => \(a = (29 - a - b) \cos 42^\circ\) Now we can solve for one of the variables, let's solve for a in terms of b: \(a = (29 - a - b) \cos 42^\circ\) Divide by \(\cos 42^\circ\): \(a \div \cos 42^\circ = 29 - a - b\) Solve for a: \(a = \frac{29 - b}{1 + \cos 42^\circ}\) Now, substitute this into the tangent equation in Step 2: \(\tan 42^\circ = \frac{b}{\frac{29 - b}{1 + \cos 42^\circ}}\) Multiply both sides by \(\frac{29 - b}{1 + \cos 42^\circ}\): \(b \tan 42^\circ = 29 - b\) Now, solve for b: \(b (1 + \tan 42^\circ) = 29\) \(b = \frac{29}{1 + \tan 42^\circ}\) Now we have the value of b, and we can find the value of a using the equation we found in Step 5: \(a = \frac{29 - b}{1 + \cos 42^\circ}\) Finally, we can find the value of c using the perimeter equation: \(c = 29 - a - b\)

Using the equations from Steps 5 and 6, let's find the lengths of the sides: 1. \(a = \frac{29 - b}{1 + \cos 42^\circ} \approx 18.81\) 2. \(b = \frac{29}{1 + \tan 42^\circ} \approx 11.58\) 3. \(c = 29 - a - b \approx 9.39\) So, the lengths of the sides of the right triangle are approximately a ≈ 18.81, b ≈ 11.58, and c ≈ 9.39.