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Indirect measurement Many problems involving right triangles require more than a single calculation in order to find the desired information. Thus, we need indirect measurement. Example Determine the length of side x to the nearest tenth of a centimeter. y x 15 cm z 30° 20 ° x=y+z • Since the value of x is the sum of the lengths of two legs of two right triangles, assign variables to those two legs and find their values separately. • For this question, it does not matter which length is calculated first, while in others, the order is crucial to determine the unknown values. Similar triangles • Similar triangles are triangles in which the corresponding angles have the same measure. The corresponding sides in similar triangles are proportional. One way of constructing similar right triangles is shown in the given diagram below. B3 26 cm B2 29 cm B1 25 cm 37 cm 14 cm A Compass and Bearings • Compass directions may be given as north (N), south (S), east (E), and west (W); or, halfway between these, such as northwest (NW). Directions can also be given as the number of degrees east or west, north or south. This is called a heading. • For headings, north or south is listed first. For example, a heading of N40ºE means 40º east of due north, as shown in this illustration. 40 W ° N40°E E • A bearing is a three-digit number of degrees between 000º and 360º that indicates a direction measured clockwise from north. For example, a bearing of 127º is in a direction 127º measured clockwise from due north, as shown below. N 127° Bearing 127° of Using Trigonometry to solve for a side Let’s review how to use trigonometry to solve for a side. In order to find an unknown side measure in a right triangle, we can use trig ratios. In addition, we must have two conditions: length of one other side and the measure of one of the acute angles. Now, let’s see an example: 62 Hypotenuse=10m Adjacent Opposite=X We should know three sides in a right triangle: hypotenuse side, adjacent side and opposite side. First, we identify the positions of the known and required side lengths of the triangle relative to the acute angle, whose measure is known. This is very important for using trigonometry to solve for a side. Since the length of the hypotenuse is known and the length of the opposite side is required, we can use sine ratio. Sine62 = opp / hyp = x / 10 X=8.8(m) Do not forget the unit! Using Trigonometry to Solve for an Angle In order to find the measure of one of the acute angles in a right triangle when the measure of each acute angle is unknown, we must know the lengths of two of the three sides. 6cm 15cm θ Here is an example: We solve forθto the nearest tenth of a degree. First, we label the given sides relative to the angle whose measure you are trying to determine. Since the lengths of the opposite side and the hypotenuse are known, we use sine ratio. sinθ= 6/15 θ= 23.6° In addition, sum of the angles in a triangle is 180°. Right Triangles • A right triangle has a right (90º) angle. The other two angles are acute (between 0º and 90º). • The side opposite the right angle is the longest side, and is called the hypotenuse. • The two sides adjacent to the right angle are called legs of the right triangle. The word adjacent means “beside”. Pythagorean Theorem In any right triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs). You can see the following illustration. A b C c a B Don’t forget: Vertices are often labeled with capital letters, and the sides opposite the vertices are labeled with the corresponding lower case letters. (You can see in the illustration) Sine, Cosine, and Tangent Ratios The three primary trigonometric ratios: Sine, Cosine, and Tangent Ratios describe the ratios of the different sides in a right triangle. • These ratios use one of the acute angles as a point of reference. The 90º angle is never used. In the following illustration, the ratios are described relative to angle θ. • Notice that the abbreviations for sine, cosine, and tangent are sin, cos, and tan. Hypotenuse Opposite side to θ Adjacent side to SOHCAHTOA Remenber!! sinθ=opposite/Hypotenuse cosθ= Adjacent/Hypotenuse tanθ= Opposite/Adjacent Angles of elevation and depression. Question: This leads out the knowledge of angles of elevation and depression. An angle of elevation is measured between two rays that have a common starting point called the vertex of the angle. One ray is horizontal, while the other is above the horizontal. An angle of depression is similar to an angle of elevation, except that the second ray is below the horizontal ray. •