2. Math 2 Honors - Santowski1 Lesson 42 - Review of Right Triangle Trigonometry Math 2 Honors – Santowski

3. Math 2 Honors - Santowski 2 (A) Review of Right Triangle Trig Trigonometry is the study and solution of Triangles. Solving a triangle means finding the value of each of its sides and angles. The following terminology and tactics will be important in the solving of triangles. Pythagorean Theorem (a 2 +b 2 =c 2 ). Only for right angle triangles Sine (sin), Cosecant (csc or 1/sin) Cosine (cos), Secant (sec or 1/cos) Tangent (tan), Cotangent (cot or 1/tan) Right/Oblique triangle

4. Math 2 Honors - Santowski 3 (A) Review of Right Triangle Trig In a right triangle, the primary trigonometric ratios (which relate pairs of sides in a ratio to a given reference angle) are as follows: sine A = opposite side/hypotenuse side & the cosecant A = cscA = h/o cosine A = adjacent side/hypotenuse side & the secant A = secA = h/a tangent A = adjacent side/opposite side & the cotangent A = cotA = a/o recall SOHCAHTOA as a way of remembering the trig. ratio and its corresponding sides

5. Math 2 Honors - Santowski 4 (B) Examples – Right Triangle Trigonometry Using the right triangle trig ratios, we can solve for unknown sides and angles: ex 1. Find a in ABC if b = 2.8, C = 90°, and A = 35° ex 2. Find A in ABC if c = 4.5 and a = 3.5 and B = 90° ex 3. Solve ABC if b = 4, a = 1.5 and B = 90°

6. Examples – Right Triangle Trigonometry 53/8/2016 Math SL1 - Santowski

7. Examples – Right Triangle Trigonometry 63/8/2016 Math SL1 - Santowski

8. Math 2 Honors - Santowski 7 (C) Cosine Law The Cosine Law states the following: a² = b² + c² - 2bccosA b 2 = a 2 + c 2 - 2accosB c 2 = a 2 + b 2 - 2abcosC We can use the Cosine Law to work in right and non-right triangles (oblique) in which we know all three sides (SSS) and one in which we know two sides plus the contained angle (SAS). a c b A B C

9. Math 2 Honors - Santowski 8 (D) Law of Cosines: Have: two sides, included angle Solve for: missing side ( missing side ) 2 = ( one side ) 2 + ( other side ) 2 – 2 ( one side )( other side ) cos( included angle ) c 2 = a 2 + b 2 – 2 a b cos C C c A a b B

10. Math 2 Honors - Santowski 9 (D) Law of Cosines: C c A a b B a 2 + b 2 – c 2 2ab cos C = Have: three sides Solve for: missing angle Missing Angle Side Opposite Missing Angle

11. Math 2 Honors - Santowski 10 a=2.4 c=5.2 b=3.5 A B C (E) Cosine Law - Examples Solve this triangle

12. Math 2 Honors - Santowski 11 (F) Examples Cosine Law We can use these new trigonometric relationships in solving for unknown sides and angles in acute triangles: ex 1. Find c in CDE if C = 56°, d = 4.7 and e = 8.5 ex 2. Find G in GHJ if h = 5.9, g = 9.2 and j = 8.1 ex 3. Solve CDE if D = 49°, e = 3.7 and c = 5.1

13. 12 (G) Review of the Sine Law If we have a non right triangle, we cannot use the primary trig ratios, so we must explore new trigonometric relationships. One such relationship is called the Sine Law which states the following: 3/8/2016 Math 2 Honors - Santowski

14. 13 (G) Law of Sines: Solve for Sides C c A a b B Have: two angles, one side opposite one of the given angles Solve for: missing side opposite the other given angle Missing Side a sin A = b sin B 3/8/2016 Math 2 Honors - Santowski

15. 14 (G) Law of Sines: Solve for Angles C c A a b B Have: two sides and one of the opposite angles Solve for: missing angle opposite the other given angle Missing Angle a sin A = b sin B 3/8/2016 Math 2 Honors - Santowski

16. 15 (H) Examples Sine Law We can use these new trigonometric relationships in solving for unknown sides and angles in acute triangles: ex 4. Find A in ABC if a = 10.4, c = 12.8 and C = 75° ex 5. Find a in ABC if A = 84°, B = 36°, and b = 3.9 ex 6. Solve EFG if E = 82°, e = 11.8, and F = 25° There is one limitation on the Sine Law, in that it can only be applied if a side and its opposite angle is known. If not, the Sine Law cannot be used. 3/8/2016 Math 2 Honors - Santowski

17. 16 (H) Homework Nelson S6.1