5.5 The Law of Sines Pre-Calculus. Use the Law of Sines to solve non-right triangles. Today we will focus on solving for the sides and angles of non-right triangles when given two angles and a side. Derivation: The Law of Sines Example : In ABC, A 49, a = 3 and B 0. Solve ABC. Example : The bearings of two ramps on the shore from a boat are 5 and 3. Assume the two ramps are 855 feet apart. How far is the boat from the nearest ramp on shore if the shore is straight and runs north-south? 5
5.6 The Law of Cosines Pre-Calculus. Use the Law of Cosines to solve non-right triangles.. Find the area of a non-right triangle when given SAS. 3. Find the area of a non-right triangle when given SSS using Heron s Formula. The Law of Sines works well when we are given AAS, ASA or when we have a magic pair. But what if we are not able to find an angle and the side across from it? What if we are given or? The Law of Cosines: a b c bccosa Note that we must know two sides to use this law. We must also know either the angle A between the two sides (SAS) or the third side a. Once a problem is started with Law of Cosines, you SHOULD continue with this law to find all missing pieces! Example : Solve WXY if x = 7cm, y = 6. cm and W = 50. Area of a Non-Right Triangle: Area ab sin C Example : Find the area of ABC when A 49, c = 3 and b = 8. While Area ab sin C works to find the area when given SAS what if we are given SSS? Heron s Formula: When given a, b, and c in a non-right triangle Semiperemeter : S ab c Area s s as bs c Example 3: Bob wants to sod a portion of his backyard roughly in the shape of a triangle with sides 9 feet, feet and 5 feet. How many 4.5 square foot sod rolls does Bob need to buy? 5
5-5 The Law of Sines: Ambiguous Case Pre Calculus. Determine when a triangle is not possible using Geometry and the Law of Sines.. Determine when a situation yields two triangles and solve for BOTH triangles using the Law of Sines. So, what about SSA? We never used SSA in Geometry! Remember AAS, ASA, SAS, and SSS were theorems in Geometry because those pieces always created two congruent Triangles. Use the given pieces of a triangle (already drawn below) and a ruler to try and create triangles. Notice one side is dashed because its length is unknown. AB = 4.5 cm A 30 BC = 3 cm B A The problem with Two Sides and a NON-Included Angle is there is more than one possibility for your answers! We call it the ambiguous case. No s: : s: Example : Determine from the given information if zero, one or two triangles may exist. Explain! a) W = 56, w = 30, x = 6 b) R = 5, r = 6, s = 4 c) A = 38, b =, a = 4 Example : Solve the triangle in Example 3c. 5 3
Fundamental Identities Day Pre-Calculus. Know the fundamental identities: reciprocal, quotient, Pythagorean, cofunction and even-odd.. Rewrite trigonometric expressions using the following techniques: a. Rewrite with sine and/or cosine b. Use the fundamental identities listed in target # c. Factor with the GCF 3. PROVE trigonometric identities using all of the above. For more background information, read the Chapter 5 Overview (pgs 443-444) This chapter is particularly important for those continuing on into college mathematics. We are now shifting our focus from problem solving to theory and proof. We are studying the connections among the trigonometric functions themselves. Identities: Mathematical sentences that are true for all values of the variable for which both sides of the equation are defined. Truth statements. Similar to postulates and theorems in Geometry. Reciprocal Identities: csc sin sin csc sec cos cos sec cot tan tan cot Quotient Identities: sin tan cos cos cot sin We already know these Identities, but are there others? Since we began our understanding of Trigonometry with the Unit Circle and right triangles, let s return there and look for patterns.. Can we relate x & y?. How are the two acute angles related? 5 4
Pythagorean Identities: cos sin Cofunction Identities (aka Complementary Angles): sin tan sec cos cot csc Odd-Even Identities: sin csc cos sec tan cot To simplify or rewrite trigonometric expressions: The final answer should be simpler than the original expression. Prefer one trig. function. Prefer NO fractions. If you are stuck try the following strategies (more strategies in the next lesson) I. By changing to sines and cosines... II. By using Pythagorean and/or Co-Function identities.... cot sec sin. sin cos cos 5 5
III. By factoring the GCF... IV. By using Even-Odd and Reciprocal identities 3. cos cossin 4. sin( x)csc( x) Proving Identities From Geometry, a proof is a series of statements, facts, definitions, postulates, etc organized in a logical order to deductively reason that a conclusion follows from a given statement(s). In Trigonometric proofs, we do the same thing using identities but we organize our proof as a series of algebraic manipulations that are sufficiently obvious enough to require no additional justifications according to your book. In other words, start with the most complex side of the equation and use identities to write equivalent statements until you get to the other side of the original equation. You will use some of the same strategies we used in simplifying. Prove each identity. 5. csc cot cot 6. sec sec x cos x tan x 5 6
Fundamental Identities Day Pre-Calculus. Rewrite trigonometric expressions using the following additional techniques: d. Combine with Least Common Denominator (LCD) e. Factor with Difference of Squares f. Factor using trinomials (the box). PROVE trigonometric identities using all of the above. V. By combining with LCD... 7. cos x cos x 8. sec x sin x sin x cosx VI. By Factoring with the Difference of Squares... 9. cos cos 0. sec x tan x VII. By Factoring with trinomials with a box... 4. cos x sin x cos x sin x cos x Remember you can use any of the above strategies and/or any of the strategies from day to PROVE! Prove each identity. cos. sin sin sec cos 5 7
5.3 Sum and Difference Identities Pre-Calculus. Use the sum and difference identities to prove other identities.. Use the sum and difference identities to rewrite an expression as the sine, cosine and/or tangent of a single angle. 3. Use the sum and difference identities to find the exact value of an expression. Sum and Difference Formulas sin ab sin acos b cos asin b cos ab cos acos b sin asin b sin ab cos ab Remember:. Sine Sign.. Cosine Sign. tan tan a tan b ab 3. Tangent. tanatanb tan ab Example : Prove each of the following is an identity. a) cos cos b) sin cos Example : Write the expression as the sine, cosine or tangent of a single angle. Then, evaluate if possible. a) sin 95 cos 55 + cos 95 sin 55 b) cos cos sin sin 4 3 4 3 c) tan 0tan 34 tan0tan34 5 8
Example 3: Use the sum and difference identities to find the exact value of each function. a) sin (5) b) cos (65) c) 5 sin d) tan 5 5 9
5.4 Multiple Angle Identities Pre-Calculus. Use the double angle identities to prove other identities.. Use the double angle identities to rewrite an expression as the sine, cosine and/or tangent of a single angle. 3. Use the double angle identities to find the exact value of an expression. While there are several identities in this section, we will only concern ourselves with the Double Angle identities which come straight from the Angle Sum Identities. Derivation : Using sin( ) sin cos cossin Derivation : Using cos( ) coscos sinsin Or using the Pythagorean Identity: cos x sin x Derivation 3: Using tan tan tan( ) tan tan Example : Verify the following identities: a) cos sin cot b) sin tan tan 5 0
Example : Write as the function of one angle. Simplify, if you can, without using a calculator. a) sin5cos5 b) cos 8 c) sin 0 d) tan5 tan 5 Example 3: If sin A, and angle A is in the first quadrant, determine: 3 a) cos A b) tan A 5
Solving Trigonometric Equations Pre-Calculus. Solve trigonometric equations using identities.. Find the correct solution(s) using the given domain and/or inverse notation. To Solve Trig Equations:. Use identities to get one trigonometric function, if possible.. Identify your variable. If x only OR x only isolate x. If x and x factor so you can use the Zero Product Property to isolate x. 3. Use the inverse to find the missing angle. 4. Give the correct solution(s) based on the directions and context of the problems. Example : Solve each equation for [0, π). a) tan x tan x 0 b) sin x 5sin x 0 c) sin x cos x Example : Find all real solutions. a) sin x 0 b) cos cos cos x x x c) 4sin x4sin x 0 5