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