math analysis[1][2]
DESCRIPTION
TRANSCRIPT
By: Krista Higgins &
Holly Maurer
The unit circle is the key tool in trig. It is beneficial to memorize the circle or be able to derive the points needed.
What is a Radian? A radian is another way to measure an angle. Not in degrees, but are listed on the unit circle using .
Radian is when the radius is equal to the arc length.
There are 2 radians in one revolution of a circle.
To convert degrees to radians use degree(/180°)
Example: Convert 240° to radians.Solution: 240°(/180)= 4/3
To convert degrees to radians use radians(180/)
Example: Convert 3/4Solution: 3/4(180/)= 135°
The formula for finding arc length is S=rθ Example: The minute hand on a clock is 8
inches. The minute hand moves 150°. Using radians find how far the hand moves.
Solution: S= 8(5/6) = 20.944 in.
Linear speed formula: V= s/t, s is the distance traveled, t is the time traveled, and v is the linear speed. It can also be expressed as V=r ω
Angular speed formula: ω=θ/t, θ is the angle in radians, t is the time, and ω is the angular speed.
Examples:
Linear Speed: A wheel with a radius of 15”, turns at a rate of 3rev./sec. How fast is it moving?
V= 15(6 /1) V=282.743 x 12
Solution: V=3392.920 inches/sec
Angular Speed: A wheel rotates 3 ft. at a rate of 15 rev. every 10 seconds. What it’s angular speed?
15rev.=30 ω= 30 all real from -1 to 1 inclusive /10
Solution: ω=9.425
Can use when trying to find the exact value of a trig. Function, along with using the unit circle.Sin θ=opposite/hypostasis Csc θ=
1/opposite/hypostasis Cos θ= adjacent/hypostasis Sec θ=1/
adjacent/hypostasisTan θ= opposite/adjacent Cot θ= 1/ opposite/adjacent
Example: Find the exact value of /3 =60 °
2
1
radical 3
Sin /3 = sin 60 ° = radical 3/2 Csc /3 =csc 60° =2radical3/3
Cos /3 = Cos 60 °=1/2 Sec /3 =sec60°= 2
Tan /3 = Tan 60 ° = radical 3 Cot /3 =tan60°=radical3/3
To find the value of all 6 trig. Functions you can to use the 6 theorems But only on a right triangle. Sin θ=opposite/hypostasis Csc θ= 1/opposite/hypostasis Cos θ= adjacent/hypostasis Sec θ=1/
adjacent/hypostasis Tan θ= opposite/adjacent Cot θ= 1/
opposite/adjacent
Example: Use the 6 trig.functions to finish the triangle.
C^2= A ^2 x B ^2 5^2=3^2 x B^2
25=9 x B25-9= 16
53
θ
Xtake the square root of 16, which =4 so X =4
Sin θ= 3/5 Csc θ=5/3Cos θ θ = 4/5 Sec θ=5/4Tan θ=3/4 Cot θ=4/3
Domain RangeSin all reals # all real from -1 to
1 inclusive
Cos all reals # all real from -1 to 1 inclusive\
Tan all real # except odd all real #
multiples of /2
Csc all real # except all real # 1> θ< -1
integral of
Sec all real # except odd all real # 1> θ< -1
multiples of /2
Cot all real # except
integral of
You can use these three identities to solve for Trig. Functions.
Sin^2 θ + Cos ^2 θ= 1 Tan ^2 θ +1= Sec ^2 θ 1+ Cot ^2 θ= Csc ^2 θ Example: If sinθ= .5, find the exact value of
cos^2θ.Solution: Sin^2θ + Cos^2θ=1
.5^2+ Cos^2θ=1 .25 + Cos^2θ=1 -.25 -.25
Cos^2θ= .75
When graphing sine or cosine remember the period is 2/k.
And the tangent period is just .
When you have 2Cos1x. 2 is the amplitude and the 1 is the period used.
When using inverses, to make them exist you have to make a domain restriction Example: - /2 < x> /2
Example: Find exact value of sin^-1(radical3/2) Solution: /3