Wrapping Up Trig Flash Cards!

Converting between Degrees and Radians

360°=2π radians 180°= π radians 1 radian= 180°/π In order to convert from degrees to radians you must multiply the degrees by π/180. Meanwhile, if you want to convert from radians to degrees, you must multiply radians by 180/π (this is to cancel out the π in the radians). Additionally, you multiply by π/180 because 2π radians are in 360 degrees so 2π/360 = π/180.

Verifying Identities 2 Verify the identity sec(x)-tan(x)sin(x)=1/sec(x)

Because the left side is going to be much easier to manipulate, we will work on the left side. Let's rewrite the functions on the left side in terms of sin(x) and cos(x) giving us 1/cos(x) - sin(x)/cos(x) * sin(x) Then we can multiply the terms to get 1/cos(x) - sin^2(x)/cos(x) Subtracting the terms gives us (1-sin^2(x))/cos(x) Pythagorean theorem says (sin^2(x) + cos^2(x) = 1) which you can manipulate cos^2(x) = 1 - sin^2(x) Substituting this gives us cos^2(x)/cos(x) = cos(x) We can rewrite cos(x) as 1/sec(x) therefore 1/sec(x) = 1/sec(x).

Arc Length 2 A pizza has a diameter of 10. Additionally, Keerat ate a portion of the pizza with an angle of 45 degrees. What is the arclength of the portion of pizza Keerat ate?

First of all we need to find circumference. Since we were given diameter we can find it from C = πd = π(10) = 10π Then we need to find the percentage of pizza Keerat ate. 45/360 = 1/8 Finally multiplying both together gives us 10π * 1/8 = 10π/8 = 5π/4.

Law of Sines 2 Given a triangle ABC where b has side length 15, C has angle 45, and B has angle 70. What is the length of side c?

First we are given AAS. Now we can set up the proportion for b/sinB = c/sinC. Then plugging in the numbers we get 15/sin70 = c/sin45. Then we solve and we eventually get c = 16.49.

Evaluating 6 trig functions at unit circle points 2 Evaluate the 6 trig functions at the 45 degree unit circle angle.

First we know that 45 degrees intersects the unit circle at (sqrt(2)/2, sqrt(2)/2). Sine(45) = y-coordinate or sqrt(2)/2 Cosine(45) = x-coordinate or sqrt(2)/2 Tangent(45) = y/x = (sqrt(2)/2)/(sqrt(2)/2) = 1 Cosecant(45) = 1/y = 1/(sqrt(2)/2) = sqrt(2) Secant(45) = 1/x = 1/(sqrt(2)/2) = sqrt(2) Cotangent(45) = x/y = (sqrt(2)/2)/(sqrt(2)/2) = 1

Evaluating Inverse Trig Functions 2 Find the answer to arccos(sqrt(3)/2))

In order to solve this problem, we need to think cosine of what angle is going to give us sqrt(3)/2. When we look on the unit circle we see that cos(30) and cos(330) both equal sqrt(3)/2. However, the restricted domain for arccos is from 0 to pi. This is because any other angles will not make arccos a function when graphed. Therefore 30 degrees fits this restriction and the answer is 30 degrees.

Evaluating 6 trig functions at unit circle points

Sine = the y-coordinate of the unit circle coordinate Cosine = the x-coordinate of the unit circle coordinate Tangent = Sine(x)/Cosine(x) or y-coordiante/x-coordinate Cosecant = 1/Sine(x) Secant = 1/Cos(x) Cotangent = Cosine(x)/Sine(x)

Arc Length

The arc length is the portion of the circumference of a circle. In order to find the portion of the circumference that you want, you need to first find the circumference. C = 2πr To find the fraction of the circle you want you take the ANGLE/360 degrees. This is because the entire circle has 360 degrees. In other words the formula for arc length is ARC LENGTH = C * ANGLE/360.

Law of Sines

This is the formula for law of sines: a/sinA = b/sinB = c/sinC. The law of sines is used when you are given a triangle with ASA or AAS. The lowercase letters in the formula are the sides while upercase letters are the angles. The side a is opposite the angle A and same with other sides and angles.

Coterminal angles 2 Keerat wants to find at least 3 examples of angles that are coterminal to the angle 45.

To find 3 angles, we need to first see where it is. Drawing 45 degrees shows us it has terminal side in quadrant 1. We know that coterminal angles are multiples of 360 degrees from the current angle. Therefore, we can add 360 degrees or subtract 360 degrees to find coterminal angles. 1. 45 + 360 = 405 degrees 2. 45 - 360 = -315 degrees 3. 45 + 360(2) = 765 degrees

Coterminal angles

Two angles in standard position that have the same terminal side. An example of coterminal angles is 30 degrees and 390 degrees. 30 degrees has terminal side in quadrant 1. 390 degrees goes around the unit circle once (circle has 360 degrees) and 30 more degrees into quadrant 1. Therefore, they are coterminal angles.

Converting between Degrees and Radians 2 Convert 135 degrees to radians.

When converting degrees to radians, we multiply by π/180. Therefore, in this case we take 135 * π/180 => 135π/180. Simplifying the numerator and denominator gives us 3π/4.

Evaluating Inverse Trig Functions

When evaluating inverse trig functions there are arcsin, arccos, arctan, arccsc, arcsec, and arccot. Inverse functions take in a number in that respective functions domain, and outputs an angle. For example the arcsin(1/2). First of all 1/2 fits the domain for arcsin which is between -1 to 1. Now, think about the sine of what angle is equal to 1/2. If we think of the unit circle sine of 45 degrees is 1/2. Therefore the answer is 45 degrees.

Verifying Identities

When given an identity with two expressions on both sides of the equal sign, you must verify it. First pick a side to work on (usually the longer or harder side so that you can simplify it to the smaller side). Then try to get the same amount of terms as the side you are verifying it with. Additionally, the Pythagorean theorem (sin^2(x) + cos^2(x) = 1) can be used to help.