Additional Topics in Trigonometry

Use Law of Sines and the Law of Cosines to solve oblique trianglesFind areas of Oblique triangles

Represent vectors as directed line segmentsPerform mathematical operations on vectors

Find direction angles of vectorsFind the dot product of two vectors and use properties of the dot product

Multiply and divide complex numbers written in trigonometric formFind powers and nth roots of complex numbers

Law of Sines𝑎sin 𝐴=

𝑏sin𝐵=

𝑐sin𝐶

Oblique TrianglesC

cA

a

B

b h

A is acute

C

cA B

bah

A is obtuse

For the trianlge

Given Two Angles and One Side -- AAS

Find the remaining angle and sides.

A pole tilts toward the sun at an 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 . How tall is the pole?

Given Two Angles and One Side -- ASA

For the triangle a=22 inches, b=12 inches, and A=. Find the remaining side and angles.

Single Solution Case -- SSA

𝑎=15 ,𝑏=25 ,𝑎𝑛𝑑 𝐴=85 °No-Solution Case -- SSA

This contradicts the fact that . So, no triangle can be formed having sides a=15 and b=25 and an angle of

𝑎=12𝑚𝑒𝑡𝑒𝑟𝑠 ,𝑏=31𝑚𝑒𝑡𝑒𝑟𝑠 ,𝑎𝑛𝑑 𝐴=20.5 °Two-Solution Case --SSA

, you can conclude that there are two possible triangles (because h<a<b).

There are two angles . Find all measures of both angels

The area of any triangle is one-half the product of the lengths of two sides times the sine of their included angle. That is, Area of an Oblique Triangle

Find the area of a triangular lot having two sides of lengths 90 meters and 52 meters and an included angle of .

Finding the Area of an Oblique Triangle

The course for a boat race starts at point A and proceeds in the direction S to point B, then in the direction S E to point C, and finally back to A. Point C, lies 8 kilometers directly south of point A. Approximate the total distance of the race course.

An application of the Law of Sines

Larson p397

Law of Cosines

𝑏2=𝑎2+𝑐2−2𝑎𝑐 cos𝐵𝑐2=𝑎2+𝑏2−2𝑎𝑏cos𝐶

Given any triangle with sides of lengths, a, b, and c, the area of the triangle is given by

Where

Heron’s Area Formula