9.1 Trig Functions of Acute Angles
Defs of sin q, cos q, tan q;
Functions of (depend on) q (independent of the size of the triangle)
Ex 1: Find trig values (ratios) given all sides of right triangle
Ex 2: Use Pythag thm to figure 3rd side, then find trig ratios
Text example: all-integer sides (called "Pythagorean triples")
More common: at least one side is a square root
Class example: adj=8, hyp=8, so opp=7*sqrt(2)
ALGEBRA REVIEW: Section 0.3 (simplifying, rationalizing roots)
Ex 3-4 Special triangles: 45-45-90, 30-60-90
So: memorize exact values for 30, 45, 60 degrees, for sine, cosine, tangent
(also include 0 and 90)
Ex 5-6 Use calculator to find trig values/ratios -- degree (vs radian) mode
Ex 7 Use inverse sin, cos, tan to find the angle
9.2 Applications
Ex 1 What does it mean to solve a triangle? Find all the angles and all the sides.
-- draw the picture
-- find the right angle
-- label the known sides/angles and choose a variable for an unknown element
-- find a trig function using your variable and two known elements
-- substitute into the trig equation and solve
To find 3rd side, text uses Pythagorean Theorem; you could also use a different trig function.
For the angles, remember that the total of all the angles in a triangle is 180o.
Ex 2-3 Applications: usually require you to find only one of the missing elements
-- draw a picture
-- find a right triangle
-- label the known sides/angles and choose a variable for an unknown element
-- find a trig function using your variable and two known elements
-- substitute into the trig equation and solve
Ex 4 Extension ladder example: find angle if ratio horizontal:slant is to be 1:4
What trig function?
Ex 5-6 Terminology: angle of depression / angle of elevation [Ex 3 used angle of elevation]
Note in Ex 6 you can easily find either acute angle in the obvious right triangle
What trig function would you use in each case?
Ex 7-8 Height of flagpole on top of building, where the distance to the base is known:
Two triangles are involved.
You often have simultaneous equations to solve in problems like this
Where are the two triangles in a problem like #43?
9.3 Law of Cosines
Standard position for an angle:
Initial side on the positive x-axis
Angle is measured as you rotate counterclockwise until you reach the terminal side
x,y,r definitions
For angles/triangles in standard position
"opposite" becomes y,
"adjacent" becomes x,
"hypotenuse" becomes r (we picture the terminal side as the radius of a circle)
Redefine the sine, cosine, and tangent functions as:
sin θ = y/r; cos θ = x/r; tan θ = y/x
Now we can find the trig function values for 0o, 90o, as well as any angle up to 180o
(later, we'll extend this to angles over 180o as well as negative angles)
Exact values in QII:
120o works like 60o, but the sign of x has changed, so cos θ and tan θ change to negative
135o works like 45o
150o works like 30o
Law of Cosines: c2 = a2 + b2 − 2 a b cos C Note the adjustment to the Pythagorean Theorem
Any pattern of letters is OK, if you match the side (on the left) with the angle of the cosine
SAS and SSS cases -- OK for the law of cosines
Ex: solve for an obtuse angle: SAS: Example X44
Ex: solve for a side: SSS: Example X32
Caution: order of operations
9.4 Law of Sines
Statement of the law: a/sinA = b/sinB = c/sinC
Proof: for triangle ABC with altitude h from angle B to side b, h = a sinC = c sinA
Two angles given ("AAS" but any sequence of sides and angles is OK)
Unique solution; start by calculating the third angle.
Ex: X40
SSA -- ambiguous case -. See cases on pg 716
(also known as ASS -- what you feel like if you miss one of the solutions)
Explain why it's ambiguous:
no solution (hanging side too short)
vs 1 sol (right triangle)
vs 2 sol (hanging side cuts horiz side twice)
vs 1 sol (hanging side longer than given side, therefore too long to cut horiz side twice
Fact: angles in QII have the same sines as those in QI: sin(180-q) = sin(q): same y and r
Consequence: every time you calculate an angle with the law of sines, you must test
whether q or 180-q (or both or neither) will give you real triangles (sum = 180)
Ex: X15 has two solutions
Ex: X13 has no solutions
SSS: Start with law of cosines, finish with law of sines.
Suggestion: solve for the largest angle first (opposite the largest side)
This avoids the ambiguous case with the law of sines: the other angles have to be acute
Ex: like #21 Bonus Q: How do you know the three sides will really form a triangle?
(If you take sides of 6, 8, and 16, there is no triangle. Why not?)
Return to: Merced College; Don Power Updated 08/30/05 by Don Power