Types of Angles acute right obtuse straight Types of Triangles acute right obtuse hypotenuse legs

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1 MTH 065 Class Notes Lecture 18 (4.5 and 4.6) Lesson 4.5: Triangles and the Pythagorean Theorem Types of Triangles Triangles can be classified either by their sides or by their angles. Types of Angles An acute angle is one whose measure is between 0º and 90 º. A right angle is one whose measure is exactly 90 º. An obtuse angle is one whose measure is between 90 º and 180 º. A straight angle is one whose measure is exactly 180 º. The sum of the three angles of a triangle must be 180 º. Types of Triangles An acute triangle is a triangle with three acute angles. A right triangle is a triangle with one right angle. An obtuse triangle is a triangle with one obtuse angle. Practice 1 Identify the following triangles as right, obtuse or acute. Use a protractor to measure the angles when necessary. is a right triangle because is a right angle. is an obtuse triangle because is an obtuse angle. is an acute triangle because all of the angles are less than 90. Pythagorean Theorem For the remainder of this section we are going to focus on right triangles. The longest side of a right triangle is called the hypotenuse. The hypotenuse is always opposite the right angle. The other two sides are called the legs of the right triangle.

2 Pythagorean Theorem If a triangle is a right triangle with legs of length a and b, and a hypotenuse of length c, then. B c a A b C Practice 2 Find the length of the unknown side in the given right triangle. If necessary, round to 3 decimal places. The hypotenuse is 5.5 cm. This creates the following: a = 4.5, b = c, c = 5.5 Converse of the Pythagorean Theorem If a triangle has sides of length, a, b, c and, then the triangle is a right triangle, and side c is the hypotenuse. Practice 3 Determine whether the given lengths of segments can form a right triangle. a) 15, 12, 9 a = 9, b = 12, c = 15 Yes, these lengths form a right triangle.

3 b) 4 a =, b =, c = 8 Yes, these lengths form a right triangle. c) 5, 5, 10 a =, b =, c = 10 No, these lengths do not form a right triangle. Distance Formula We are now ready to learn how to find the distance between any two points in the rectangular coordinate system. We will use the Pythagorean Theorem to help us find that distance. We can find the lengths of each leg of the right triangle by taking the difference of coordinate that is not shared by those endpoints. For example, with the above triangle, we can find the length of AB, by looking at the coordinates of A and B. A, (2, 5), and B, (10, 5) give us a difference of 8 since 10 2 = 8. We can use this same argument to obtain the length of BC = 6. We can now find the distance from A to C by using the Pythagorean Theorem. Note: We do not need to consider -10 as we are talking about length and cannot have a negative length. We can generalize this process in what is called the Distance Formula. Remember that we have our and

4 Distance Formula The distance between two points and, is Practice 4 Find the distance between (4,-5) and (-2,-9). Give the answer both as an exact value and as an approximate value rounded to 3 decimal places. and Using Triangles to Solve Problems There are many applications that use triangles. We will look at some in this section and again in section 4.6. Practice 5 Richard is going to attach 3 guy wires to his CB antenna at a point 30 feet above the ground. According to the manufacturer of the antenna, the wires should be secured at ground level 17 feet from the base of the antenna. Richard wants to know how much wire he is going to need to secure his antenna. Round the answer to the nearest tenth of a foot. (Assume the antenna is located on level ground.) a =, b = With three guy wires needed for this project, this means Richard will need a total of 34.5(3) = ft. Lesson 4.6: Sine, Cosine, and Tangent Right triangle trigonometry is a branch of mathematics that deals with properties of triangles. In this section we will study one concept from trigonometry: the sine, cosine, and tangent ratios as they apply to a right triangle. The Ratios A ratio is a way of comparing two things. In this case, the lengths of the sides of a triangle. Let s start by identifying the parts of a right triangle. The best way to do this is through the perspective of one of the acute angles.

5 This is defined from the perspective of B. From that perspective, AC is considered the opposite side or leg, AB is considered the hypotenuse as it is opposite from the 90 angle, and BC is considered adjacent side or leg because it is next to B. With these sides established we are ready to write down the trigonometric ratios of angle B. The sine of B: The cosine of B: The tangent of B: Trigonometric Ratios Note: The opposite leg and adjacent leg change depending on which acute angle we are talking about. For example, if we were to look from the perspective of A, then the opposite leg and adjacent leg would change. Practice 1 For the triangle in Figure 4 write: (a) the sine, cosine, and tangent ratios of A (b) the sine, cosine, and tangent ratios of B. (a)

6 (b) Finding the Sine, Cosine, and Tangent of an Angle on a Calculator If we don't know the lengths of the sides in the right triangle, but we know the measure of the angle, we can use a calculator to find the trigonometric ratio for that angle. We will be working with angles measured in degrees. We need to make sure our calculator is in the degree mode. For a TI 83/84, to do this press the MODE key. The 3 rd line has two options: Radian and Degree. If your calculator has Radian selected, you need to select Degree. Practice 2 Using a calculator, evaluate each trigonometric function. Round answers to four decimal places where appropriate. (a) (b) (c) Finding a Missing Length in a Triangle In a right triangle we know the measure of one of the angles is 90º. If we know the length of one of the sides and we know the measure of one of the angles (other than the right angle), we can find the lengths of the other sides. Practice 3 A. A right triangle is drawn below. Find the length of side a. Round the answer to two decimal places.

7 B. A triangle is drawn below. Find the length of side b. Round the answer to one decimal place. Finding the Height or a Distance Many kinds of problems can be solved using trigonometry, from finding the height of a tree to measuring the distance across a canyon. Practice 4 (a) A 24-foot ladder leaning against a vertical wall forms an angle of 65º with the ground. To the nearest tenth of a foot, how far is the base of the ladder from the wall? 24 ft. x 65

8 (b) Steven rides his bicycle home from school every day. He normally rides 0.6 miles along Arnold Way, then turns the corner (90º) onto Willamette Avenue and rides a short distance to his home. If he could ride in a straight line from his school to his home that line would form a 23º angle with his route along Arnold Way. How far from the corner of Arnold Way and Willamette Avenue does Steven live? Round your answer to one decimal place. School Arnold Way mil. x Willamette Avenue Home