In Exercises 1-12 , solve the right triangle shown in the figure. Round lengths to two decimal p

In Exercises 1-12 , solve the right triangle shown in the...

Key Concepts

Right Triangle

A right triangle is a triangle that includes a 90° angle. The properties of right triangles allow the use of trigonometric ratios and the Pythagorean theorem to find unknown sides and angles, making them a fundamental part of geometry and trigonometry.

Trigonometric Ratios

Trigonometric ratios, including sine, cosine, and tangent, are functions that relate the angles of a triangle to the ratios of its sides. In the context of right triangles, these ratios are essential for calculating unknown side lengths and angles when one side length and one non-right angle are known.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is critical for finding the length of one side when the two others are known.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsine, arccosine, and arctangent, are used to determine an angle when given a trigonometric ratio. In solving right triangles, these functions help convert the known ratios into angle measures, completing the solution of the triangle.

Transcript

00:02 We are supposed to solve a right angle triangle from the given dimensions, and they're supposed to round off to two decimal places, that is our length and then our angles to the nearest tank.

00:16 We are given angle a is 23 .5.

00:20 So angle a is equal to 23 .5 degrees.

00:25 And we are also given side b.

00:27 Side b is equals to 10.

00:30 So we are supposed to solve the rest of the triangle.

00:35 Now, firstly do a sketch so that you have got a picture of what you are solving.

00:44 So what we have is a here, b and c is always the right angle side.

00:52 Then a is given as 23 .5 and b is 10.

01:00 So what we are given is a side that is adjacent to the given angle.

01:07 So we want to calculate the rest of the angle.

01:10 So firstly, just solve the angle problem.

01:14 We know that if it's a right angle triangle, the other two sides, other than see, the other two sides, which are not 90 degrees, are complementary.

01:24 So they add up to 90 degrees.

01:27 So the other two sides are complementary.

01:31 They add up to 90 degrees.

01:33 Therefore, if you want to find one, if it's missing, just say 90 degrees minus that angle, which is given and you should be able to get the missing angle.

01:47 So in this case, it's 90 degrees minus 23 .5, which is 66 .5.

02:04 Degrees.

02:06 So the angle is done.

02:07 Now we are left with the other two sides that are missing.

02:11 All right, we can start with maybe side a.

02:14 Side a is opposite to the given adjacent side.

02:19 So we can use two conometric ratio to solve it...