Test 11(trig)

How to solve: If cot(x-10)=tan(4x) the value of x is...

tanϴ and cotϴ must add to 90 degrees, so (x-10)+(4x)=90 and solve. The answer is 20.

Tangent and cotangent are cofunctions

tanϴ=cot(90-ϴ)

Tan(-110) as a positive acute angle!

-110+360=250 250-180=70

Theta Symbol

ϴ

Reference angles in Q3

ϴ-180=reference angle

Reference Angles in Q1

They are there own reference angles!

Pythagorean Theorem

Used to solve for a side!

Sinϴ=

Vertical Distance

Terminal side

Where the rotation ends

Angles in the standard position are classified by...

Where the terminal side is.

Cosine

adjacent/hypotenuse

Coterminal Angles and how to find them

-Angles in standard position that share the same terminal side. -To find these angles, add and subtract 360. -Ex: Coterminal angles of 300 are -60, 660, etc...

How do you find values using a unit circle if you are not given an angle of 45, 30, or 60?

-Graph the angle/triangle -Find the reference angle -Use sine/cosine/tangent of ϴ

Reference Triangles

-Used to find trigonometric values for their standard position angles whose terminal sides reside in Quadrants II, III and IIII

Reference Angles in Q2

180-ϴ=reference angle

How many radians is 180 degrees? 360? 90? 270?

180=π 360=2π 90=π/2 270=3π/2

The pendulum of a clock swings at an angle of 2.5 radians as its tip travels through an arc of 50 centimeters. Find the length of the pendulum, in centimeters.

20cm

Through how many radians does the minute hand of a clock turn in 24 minutes?

24/60=x/360 x=144 144/x=180/π x=.8π

Reference angles in Q4

360-ϴ=reference angle

If you have a point (4, 3), where is it on a graph?

4 to the right(cosϴ) and 3 up(sinϴ).

Sin90= Cos90= Tan90=

90 is at the point (0, 1) Sin90=1 Cos90=0 Tan90=undefined

A large pizza has a diameter of 18 inches. A pizza is typically cut into 8 slices. What is the length of a piece of crust of one slice of pizza?

9π/4

Confunctions

A function of any angle is equal to the cofunction of its complement.

A circle has a radius of 5 degrees. Find the length of the arc intercepted by an angle of 100 degrees.

Convert 100 to radians=5π/9 radians. 5π/9=x/5 x=25π/9

Secant

Cosine reciprocal: hypotenuse/adjacent

Cosϴ=

Horizontal Distance

Radian Measure Formula

Length of intercepted arc/length of radius=measure of angle in radians.

Quadrantal Angles

Multiples of 90(90, 180, 270, 360)

Initial side

Positive x axis where oration begins

Amary Seck Teaches Chem

Q1: All positive Q2: Only sine(and cosecant) positive Q3: Only tangent(and cotangent) positive Q4: Only cosine(and secant) positive

Sin180= Cos180= Tan180=

Sin180=0 Cos180=-1 Tan180=0

The Unit Circle 30-60-90

Sin30=1/2 Cos30=root(3)/2 Tan30=1/root(3) Sin60=root(3)/2 Cos60=1/2 Tan60=root(3)

The Unit Circle 45-45-90

Sin45=root(2)/2 Cos45=root(2)/2 Tan45=1

Cosecant

Sine reciprocal: hypotenuse/opposite

The Unit Circle

Sinϴ=y/1=y (directed vertical distance) Cosϴ=x/1=x (directed horizontal distance) Tanϴ=y/x=sinϴ/cosϴ The point (x, y) is equivalent to (cosϴ, sinϴ)

REMEMBER

TO CHECK IF A VALUE IS NEGATIVE OR POSITIVE NO MATTER WHAT YOU'RE DOING!

Cotangent

Tangent reciprocal: adjacent/opposite

Reference Angles

The reference angle is an acute angle formed by the terminal side of the given angle and the x-axis. Reference angles may appear in all four quadrants.

Cos120 as a positive acute angle!

cos60(make triangle, solve)

Cos280 as a positive acute angle!

cos80

Solving for a side given an angle and 1 side.

ex: tan67=x/14 tan67*14=x

Converting degrees to radians and radians to degrees

ex: 120 degrees 120/x=180/π x=2π/3 ex: 2π/3 2π/3/x=π/180 x=120

Recall: Arc length

length of intercepted arc/length of radius=measure of angle in radians

Tangent

opposite/adjacent

Sine

opposite/hypotenuse

Secant and Cosecant are cofunctions

secϴ=csc(90-ϴ)

Solving for an angle

sinx=1/4 sin-1(1/4)=x

Sine and Cosine are cofunctions

sinϴ=cos(90-ϴ)

If the coordinates of point B are (-1/2, root(3)/2), state the sinϴ, cosϴ, and tanϴ

sinϴ=root(3)/2 cosϴ=-1/2 tanϴ=-root(3)