Pre-CalculusChapter 6

Additional Topics in Trigonometry

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6.1 The Law of Sines

Objectives: Use Law of Sines to solve oblique

triangles (AAS or ASA). Use Law of Sines to solve oblique

triangles (SSA). Find areas of oblique triangles. Use Law of Sines to model & solve

real-life problems.

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Oblique Triangles Have no right angles. Angles labeled with

capital letters. Sides labeled with

lower-case letters (same letter as opposite angle).

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Proof of Law of Sines Let h be the altitude of the triangle. Find sin A and sin B.

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Proof continued…. Solve each equation for h.

Set h = h.

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The Law of Sines

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Example 1 For ABC, C = 102.3°, B = 28.7°, and b = 27.4

feet. Find the remaining angle and sides.

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Example 2 A pole tilts toward the

sun at an 8° angle from the vertical, and it casts a 22-foot shadow. The angle of elevation from the tip of the shadow to the top of the pole is 43°. How tall is the pole?

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Possible Combinations for Law of Sines

Given a, b, A, and where h = b sin A:

We will look at each one individually.

A is acutea < h a = h a > b h < a < b

A is obtusea ≤ b a > b

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A is Acute and a < h

No triangle formed - a can’t reach the base.

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A is Acute and a = h

One triangle is formed.

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A is Acute and a > b

One triangle is formed – a intersects the base at only one point.

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A is Acute and h < a < b (Ambiguous Case)

Two unique triangles can be formed. Side a can intersect the base at 2

points.

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A is Obtuse and a ≤ b

No triangle is formed – a can’t reach the base.

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A is Obtuse and a > b

One triangle is formed.

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The Ambiguous Case Given A, a, and b, we can find h

h = b sin A Is a > b? If so, only 1 triangle. Is h < a < b? If so, then 2 triangles.

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Example 3 For ABC, a = 22 inches, b = 12 inches,

and A = 42°. Find the remaining side and angles.

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Example 4 For ABC, a = 12 meters, b = 31

meters, and A = 20.5°.

a.How many triangles can be formed?

b.Find all remaining side(s) and angles.

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Example 5 Show that there is no triangle for

which a = 15, b = 25, and A = 85°.

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Area of an Oblique Triangle The area of any triangle is one-half

the product of the lengths of two sides times the sine of their included angle.

What happens if the included angle is 90°?

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Example 6 Find the area of a triangular lot

having two sides of lengths 90 meters and 52 meters and an included angle of 102°.

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Homework 6.1 Worksheet

6.1