Sines

Browse Presentation Creator Pro Upload Jul 07, 2014 150 likes | 534 Views Law of Sines. What You will learn: Use the Law of Sines to solve oblique triangles when you know two angles and one side ( AAS or ASA). Use the Law of Sines to solve oblique triangles when you know two sides and the angle opposite one of them ( SSA ). Given Two Angles and One Side – AAS. Download Presentation Law of Sines An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher. Presentation Transcript Law of Sines What You will learn: Use the Law of Sines to solve oblique triangles when you know two angles and one side (AAS or ASA). Use the Law of Sines to solve oblique triangles when you know two sides and the angle opposite one of them (SSA).Given Two Angles and One Side – AAS For the triangle below C = 102, B = 29, and b= 28 feet. Find the remaining angle and sides. By the triangle angle-sum theorem, A = B a c A C b Law of Sines Try this: By the triangle angle-sum theorem, C = Let’s look at this: Example 1 Given a triangle, demonstrate using the Law

Presentation on theme: "The Law of SINES."— Presentation transcript: 1 The Law of SINES 2 The Law of SINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles: 3 Use Law of SINES when ... AAS - 2 angles and 1 adjacent sideyou have 3 dimensions of a triangle and you need to find the other 3 dimensions - they cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation. Use the Law of Sines if you are given: AAS - 2 angles and 1 adjacent side ASA - 2 angles and their included side SSA (this is an ambiguous case) 4 Example 1 You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measures of angle C and sides b and c. * In this section, angles are named with capital letters and the side opposite an angle is named with the same lower case letter .* 5 Example 1 (con’t) A C B 70° 80° a = 12 c bThe angles in a ∆ total 180°, so angle C = 30°. Set up the Law of Sines to find side b: 6 Example 1 (con’t) A C B 70° 80° a = 12 c b = 12.6 30°Set up the Law of Sines to find side c: 7 Example 1 (solution) A C B 70° 80° a = 12 c = 6.4 b = 12.6 30°Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm Note: We used the given values of A and a in both calculations. Your answer is more accurate if you do not used rounded values in calculations. 8 Example 2 You are given a triangle, ABC, with angle C = 115°, angle B = 30° and side a = 30 cm. Find the measures of angle A and sides b and c. 9 Example 2 (con’t) To solve for the missing sides or angles, we must have an angle and opposite side to set up the first equation. We MUST find angle A first because the only side given is side a. The angles in a ∆ total 180°, so angle A = 35°. A C B 115° 30° a = 30 c b 10 Example 2 (con’t) A C B 115° a = 30 c b30° a = 30 c b 35° Set up the Law of Sines to find side b: 11 Example 2 (con’t) A C B 115° a = 30 c b = 26.230° a = 30 c b = 26.2 35° Set up the Law of Sines to find side c: 12 Example 2 (solution) Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm A115° 30° a = 30 c = 47.4 b = 26.2 35° Angle A = 35° Side b = 26.2 cm

Introduction to the Law of SinesLaw of sines may be used in the technique of triangulation to find out the unknown sides when two angles and a side are provided. It is also applicable when two sides and one unenclosed side angle are given. We may use the form to find out unknown angles in a scalene triangle. The Law of sine and cosine are the trigonometric equations that are used to calculate unknown lengths and angles for a scalene triangle.Define Law of SinesFor triangles, like we have the law of cosines, we have the law of sines. The Law of sines is a trigonometric equation where the lengths of the sides are associated with the sines of the angles related. The law of sines is described as the side length of the triangle divided by the sine of the angle opposite to the side. The formula for the sine rule of the triangle is: \[\frac{a}{sin A}\] = \[\frac{b}{sin B}\] = \[\frac{c}{sin C}\] (where a, b, c are sided lengths of the triangle and A, B, C are opposite angles to the respective sides)Therefore, side length a divided by the sine of angle A is equal to side length b divided by the sine of angle B is equal to side length c divided by the sine of angle C. It may also be written as \[\frac{a}{sin A}\] = \[\frac{b}{sin B}\] = \[\frac{c}{sin C}\] = d (where d stands for the diameter of the triangle’s circumcircle).\[\frac{sin A}{a}\] = \[\frac{sin B}{b}\] = \[\frac{sin C}{c}\] is also an acceptable form of the law of sines.Law of Sines ProofSine Rule ProofTo derive the law of sines, let us take the area of a triangle whose sides are a, b, c and the angles opposite to the respective sides are A, B, and C.[Image

Side c = 47.4 cm Note: Use the Law of Sines whenever you are given 2 angles and one side! 13 The Ambiguous Case (SSA)When given SSA (two sides and an angle that is NOT the included angle) , the situation is ambiguous. The dimensions may not form a triangle, or there may be 1 or 2 triangles with the given dimensions. We first go through a series of tests to determine how many (if any) solutions exist. 14 The Ambiguous Case (SSA)In the following examples, the given angle will always be angle A and the given sides will be sides a and b. If you are given a different set of variables, feel free to change them to simulate the steps provided here. ‘a’ - we don’t know what angle C is so we can’t draw side ‘a’ in the right position A B ? b C = ? c = ? 15 The Ambiguous Case (SSA)Situation I: Angle A is obtuse If angle A is obtuse there are TWO possibilities If a ≤ b, then a is too short to reach side c - a triangle with these dimensions is impossible. If a > b, then there is ONE triangle with these dimensions. A B ? a b C = ? c = ? A B ? a b C = ? c = ? 16 The Ambiguous Case (SSA)Situation I: Angle A is obtuse - EXAMPLE Given a triangle with angle A = 120°, side a = 22 cm and side b = 15 cm, find the other dimensions. Since a > b, these dimensions are possible. To find the missing dimensions, use the Law of Sines: A B a = 22 15 = b C c 120° 17 The Ambiguous Case (SSA)Situation I: Angle A is obtuse - EXAMPLE Angle C = 180° - 120° ° = 23.8° Use Law of Sines to find side c: A B a = 22 15 = b C c 120° 36.2° Solution: angle B = 36.2°, angle C = 23.8°, side c = 10.3 cm 18 The Ambiguous Case (SSA)Situation II: Angle A is acute If angle A is acute there are SEVERAL possibilities. Side ‘a’ may or may not be long enough to reach side ‘c’. We calculate the height of the altitude from angle C to side c to compare it with side a. A B ? b C = ? c = ? a 19 The Ambiguous Case (SSA)Situation II: Angle A is acute First, use SOH-CAH-TOA to find h: A B ? b C = ? c = ? a h Then, compare ‘h’ to sides a and b . . . 20 The Ambiguous Case (SSA)Situation II: Angle A is acute If a A B ? b C = ? c = ? a h 21 The Ambiguous Case (SSA)Situation II: Angle A is acute If h A B b C c a h A B b C c a h If we open

Oblique triangles do not have any right angles. When solving oblique triangles, we must first know the measure of at least one leg and the measure of the other two parts of the oblique triangle: two angles, two legs, or one side and one angle. In simple words, we can get a lot of different combinations when solving the oblique triangles. One of these combinations or attributes is the SAS triangle.What Is a SAS Triangle?SAS (side-angle-side) triangle is basically a triangular combination when we know the measure of two sides of a triangle and the angle between them.Consider a triangle $△ABC$ with the sides $a$, $b$, and $c$ facing the angles $\alpha$, $\beta$, and $\gamma$, respectively, as shown in Figure 15-1. We can observe that we are given two sides $b$ and $c$, and the included angle $\alpha$. Figure 14-1 illustrates a triangular combination which is known as a SAS triangle.How To Solve a SAS Triangle?When we know the measure of two sides and the included angle, we can apply a three-step method to solve a SAS triangle: using the Law of Cosines, using the Law of Sines, and determining the measure of the third angle.Step 1 of 3Use the Law of Cosines to measure the missing side.Step 2 of 3Use the Law of Sines to find the angle (acute angle) opposite the smaller of the two sides.Step 3 of 3Determine the measure of the third angle by subtracting the already measured angles (given angle and the angle determined in step 2) from $180^{\circ }$. Example 1In triangle $△ABC$, $m∠\alpha = 60^{\circ }$, $b = 2$ and $c = 3$. Solve the triangle.Solution:We are given two sides $b = 2$, $c = 3$, and an angle $m∠\alpha = 60^{\circ }$. To solve the SAS triangle, we will apply this three-step