# Math Analysis[1][2]

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## Math Analysis[1][2]Presentation 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
• Example: Convert 3/4
• Solution: 3/4(180/)= 135°
• The formula for finding arc length is S=r θ
• 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
• Example: Find the exact value of /3 =60 °
Exact Value of Trig. Functions 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
• 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 B
• 25-9= 16
Value of the 6 Trig. Functions w/ a Right Triangle 5 3 θ X
• take the square root of 16, which =4 so X =4
• Sin θ = 3/5 Csc θ =5/3
• Cos θ θ = 4/5 Sec θ =5/4
• Tan θ =3/4 Cot θ =4/3
• Domain Range
• Sin 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 
Domain and Range of Trig. Functions
• 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 sin θ = .5, find the exact value of cos^2 θ .
• Solution: Sin^2 θ + Cos^2 θ =1
• .5^2+ Cos^2 θ =1
• .25 + Cos^2 θ =1
• -.25 -.25
• Cos^2 θ = .75
Pythagorean Identity
• 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
Inverses