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)
