Trigonometry: Conversion, Identities, and Exact Values

Study Notes

The video focuses on solving basic trigonometry problems by converting degrees to radians using the formula degrees * pi / 180 to cancel out the degree symbol.
Negative angles are measured clockwise, and positive angles are measured counterclockwise.
Coterminal angles land in the same position on the graph and can be found by adding or subtracting multiples of 360 degrees.
To find the arc length, use the formula arc length = angle in radians * radius, after converting the angle from degrees to radians.
For finding the value of sine x in a right triangle, use the SOHCAHTOA rule where sine = opposite/hypotenuse.
Reciprocal trigonometric identities include secant = 1/cosine and cotangent = 1/tangent.
To find the value of secant x, first find the value of cosine using the adjacent side and hypotenuse ratio, then calculate secant as 1/cosine.- Sine x = 7/25, in quadrant one, creates a 7-24-25 triangle, allowing to find tangent x as 7/24.
In quadrant four, with tangent x = -8/15, the missing side in the triangle is 17, leading to cosecant x = -17/8.
For sine x = 5/7 in quadrant two, cosine x is found by applying the Pythagorean identity to get 2√6/7.
The quadrant in which sine z < 0 and cosine z > 0 is quadrant four.
Sine negative x = -sin x, cosine negative x = cos x, and cotangent negative x = -cotan x.
Sine pi/5 is equivalent to cosine 3pi/10.
Cosine negative 150 is equivalent to cosine 210, found by applying cofunction identity and quadrant properties.
The expression 1 + sin^2(40) + sin^2(50) can be simplified by recognizing that sin 50 = cos 40, leading to cos^2(40).
The Pythagorean identity for sine and cosine states that sine squared plus cosine squared equals one.
To find the exact value of cosine 60 without a calculator, you can use the 30-60-90 reference triangle where cosine 60 is equal to 1/2.
Another method to find trigonometric values without a calculator is by using the unit circle, where cosine 60 is also equal to 1/2.
To find the exact value of sine pi over 4, first convert the angle to degrees (45 degrees) and then use the 45-45-90 reference triangle to determine that sine 45 is equal to the square root of 2 divided by 2.
The exact value of tangent 30 can be found using the 30-60-90 triangle, where tangent 30 is equal to the square root of 3 divided by 3.
Using the unit circle, tangent 30 can also be found by dividing the y-coordinate by the x-coordinate, resulting in the same value: square root of 3 divided by 3.
The reference angle of 290 degrees can be calculated by subtracting the angle from 360 since it lies in quadrant four, resulting in a reference angle of 70 degrees.
In finding the exact value of cosine 210 without a calculator, first determine that 210 degrees is in quadrant three and calculate the reference angle as 30 degrees.
Using the 30-60-90 reference triangle, cosine 210 can be evaluated as negative square root of 3 divided by 2.

Description

Explore basic trigonometry concepts including converting degrees to radians, coterminal angles, arc length, reciprocal trigonometric identities, and finding exact values without a calculator using reference triangles and the unit circle.