angle b=25 degrees, angle c=90

AB= 25

a. Find the length of side b to two decimal places.

b. Find the length of side a to two decimal places in three different ways.

Basic application of your simple right-angled triangle properties ....

AC/25 =sin25°
AB = 25sin25 = 10.57

BC/25 = cos25°
BC = 22.66

BTW, the usual convention is to label the vertices of the triangle using capitals and label the side opposite a vertex with the corresponding small letter.
e.g. For triangle ABC
C = 90° , B = 25° and c=25

thanks! :)

a. To find the length of side b, you can use the sine function. The sine of angle B is equal to the ratio of the length of side b to the length of the hypotenuse (side AB). So, you can use the formula:

sin(B) = b/AB

Plugging in the values, we get:

sin(25) = b/25

To isolate b, we can rearrange the equation:

b = sin(25) * 25

Using a calculator, evaluate sin(25) and multiply by 25 to get the length of side b to two decimal places.

b. To find the length of side a using three different methods, you can use the following trigonometric functions:

1. Sine function: The sine of angle A is equal to the ratio of the length of side a to the length of the hypotenuse (side AB). So, you can use the formula:

sin(A) = a/AB

Substituting the known values, we get:

sin(A) = a/25

To isolate a, we rearrange the equation:

a = sin(A) * 25

Evaluate sin(A) using a calculator, and multiply by 25 to get the length of side a to two decimal places.

2. Cosine function: The cosine of angle A is equal to the ratio of the length of side a to the length of the hypotenuse (side AB). So, you can use the formula:

cos(A) = a/AB

Substituting the known values, we get:

cos(A) = a/25

To isolate a, we rearrange the equation:

a = cos(A) * 25

Evaluate cos(A) using a calculator, and multiply by 25 to get the length of side a to two decimal places.

3. Pythagorean theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (side AB)^2 is equal to the sum of the squares of the lengths of the other two sides (a^2 + b^2). So, you can use the formula:

a = √(AB^2 - b^2)

Substitute the known values, and using a calculator, evaluate √(AB^2 - b^2) to get the length of side a to two decimal places.

To find the length of side b, we can use the trigonometric function called sine (sin). In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The formula is given as:

sin(angle) = opposite/hypotenuse

In this case, we know that angle B is 25 degrees, and the hypotenuse AB is given as 25 units.

a. To find the length of side b:
Using the sine function, we have:
sin(25 degrees) = length of side b / 25
Rearranging the equation, we get:
length of side b = sin(25 degrees) * 25

Now, we can use a calculator to find the value of sin(25 degrees), and then multiply it by 25 to get the length of side b.

b. To find the length of side a, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The formula is given as:

c^2 = a^2 + b^2, where c is the hypotenuse, and a and b are the other two sides.

From the given information, side c has a length of 90 units, and side b, which we found in part a, has a length according to the calculation result. Therefore, we can rearrange the equation to solve for side a:

a^2 = c^2 - b^2
length of side a = sqrt(c^2 - b^2)

By substituting the respective values and calculating, we can find the length of side a.

Additionally, we can also use the trigonometric function called cosine (cos) to find the length of side a. In a right triangle, the cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The formula is given as:

cos(angle) = adjacent/hypotenuse

Using the value of angle B (25 degrees) and the hypotenuse AB (25 units), we can rearrange the equation to solve for the length of side a:

length of side a = cos(25 degrees) * 25

Similarly, by using a calculator to find the value of cos(25 degrees), we can multiply it by 25 to find the length of side a.

By using any of these methods, we can find the length of side a to two decimal places.