Trigonometry Law of Sines Section 6.1

Review Solve for all missing angles and sides: a 3 5 B A

Do not assume that triangles are drawn to scale.

What formulas did you use to solve the right triangle? Pythagorean Theorem SOHCAHTOA Inverse Trig function All angles add up to 180 o in a triangle What if it’s not a right triangle? GASP!! What do we do then??

Copyright © 2007 Pearson Education, Inc. Slide 10-5 Remember this… In a triangle, the sum of the interior angles is 180º. No triangle can have two obtuse angles. The height of a triangle is less than or equal to the length of two of the sides. The sine function has a range of If the θ is a positive decimal < 1, the θ can lie in the first quadrant (acute angle) or in the second quadrant (obtuse angle).

Note:  capital letters always stand for __________!  lower-case letters always stand for ________! Use the Law of Sines ONLY when:  you DON’T have a right triangle AND  you know an angle and its opposite side A B C a b c angles sides Use either equation

Derivation of the Law of Sines Start with an acute or obtuse triangle and construct the perpendicular from B to side AC. Let h be the height of this perpendicular. Then c and a are the hypotenuses of right triangle ADB and BDC, respectively.

We can use the Law of Sines to solve oblique triangles and to find the areas of oblique triangles.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Definition: Law of Sines Law of Sines If ABC is an oblique triangle with sides a, b, and c, then Acute Triangle C BA b h c a C B A b h c a Obtuse Triangle

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 The following cases are considered when solving oblique triangles using the Law of Sines. Solving Oblique Triangles 1.Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA) (Known as the Ambiguous Case) A C c A B c C c a

 The Law of Sines can be used to “ solve a triangle,” which means to find the measures of all of the angles and all of the sides of a triangle.

Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth. A 63° C a ˚79˚

Ex. 2: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth. s 40° T r ° 51˚

Ex. 3: Draw ΔABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth. A B C a. 7 37˚ 76˚ 67˚ b c

b. A B C ˚ b 14˚ 96˚

The Ambiguous Case – SSA In this case, you may have information that results in one triangle, two triangles, or no triangles.

Example #1 of SSA Two sides and an angle opposite one of the sides are given. Let’s try to solve this triangle.

By the law of sines,

Thus, Therefore, there is no value for  that exists! No triangle is possible!

Example #2 of SSA Two sides and an angle opposite one of the sides are given. Let’s try to solve this triangle.

By the law of sines,

So that, Interesting! Let’s see if one or both of these angle measures makes sense. Find the sine of both of these angles.

Case 1 Case 2 Both triangles are valid! Therefore, we have two possible cases to solve.

Finish Case 1:

Finish Case 2:

Wrapping it up, here are our two solutions:

Example #3 of SSA: Two sides and an angle opposite one of the sides are given. Let’s try to solve this triangle.

By the law of sines,

Note: Only one is legitimate!

Thus, we have only one triangle. Now let’s find b.

By the law of sines,

Finally, we have:

Trigonometry Area of a Triangle

The Area of a Triangle Using Trigonometry We can find the area of a triangle if we are given any two sides of a triangle and the measure of the included angle. (SAS)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 36 Area of an Oblique Triangle C BA b c a Find the area of the triangle. A = 74 , b = 103 inches, c = 58 inches Example 5: 74  103 in 58 in

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 37 Example 6 Finding the Area of a Triangular Lot Find the area of a triangular lot containing side lengths that measure 24 yards and 18 yards and form an angle of 80° A = ½(18)(24)sin80 A = yards

Example: Find the area of given a = 32 m, b = 9 m, and

Calculate the area of the triangle shown. Give your answer correct to one decimal place. Area of triangle = absin C 1212 Area = (3)(4) sin 55  1212 = … 4 cm 3 cm C must be the included angle = 4·9 cm 2 55º

Find the area of triangle abc, correct to the nearest whole number. Area of triangle = acsinB 1212 Area = (14)(18·4) sin 70  1212 = … 18·4 14 C AB 44º 66º C must be the included angle  ABC = 180  – 44  – 66  = 70  70º = 121units 2

The Sine Rule Application Problems 25 o 15 m A D The angle of elevation of the top of a building measured from point A is 25 o. At point D which is 15m closer to the building, the angle of elevation is 35 o Calculate the height of the building. T B Angle TDA = 145 o Angle DTA = 10 o 35 o – 35 = 145 o 180 – 170 = 10 o

The Sine Rule A The angle of elevation of the top of a column measured from point A, is 20 o. The angle of elevation of the top of the statue is 25 o. Find the height of the statue when the measurements are taken 50 m from its base 50 m Angle BCA = 70 o Angle ACT = Angle ATC = 110 o 65 o m B T C 180 – 110 = 70 o 180 – 70 = 110 o 180 – 115 = 65 o 20 o 25 o 5o5o

A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water? Answer: About 42 feet of the fishing line that is cast out is above the surface of the water.