Pre Calculus: Trigonometric functions

What is the relationship with cosine and sine?

(cos θ)^2 + (sin θ)^2 = 1

What is the area of a slice of a circle measured in radians?

A slice with angle θ radians inside a circle of radius r has area (1/2)θr^2

How do you find the length of a circular arc?

An arc corresponding to θ degrees on a circle of radius r has length θπr : 180 .

How do you convert radians to degrees or degrees to radians?

Converting radians to degrees θ radians = (180θ / π ) degrees Converting degrees to radians θ degrees◦ = θπ /180 radians

What is cosecant?

Cosecant The cosecant of an angle θ, denoted csc θ, is defined by csc θ = 1/sin θ .

What is cotangent?

Cotangent The cotangent of an angle θ, denoted cot θ, is defined by cot θ = cos θ/sin θ .

What are the dimensions of a triangle with hypotenuse equal to 1?

Dimensions of a triangle with angles of 30◦, 60◦, and 90◦ In a triangle with angles of 30◦, 60◦, and 90◦ and hypotenuse of length 1, • the side opposite the 30◦ angle has length 1/2; • the side opposite the 60◦ angle has length √3/2

What is the length of a circular arc on the unit circle?

Length of a circular arc If 0 < θ ≤ 2π, then a circular arc on the unit circle corresponding to θ radians has length θ.

What is a radian?

Radians are a unit of measurement for angles such that 2π radians correspond to a rotation through an entire circle. Angles without units: If no units are given for an angle, then assume the units are radians.

What does the function cosine represents?

The cosine of θ, denoted cos θ, is the first coordinate of the endpoint of the radius of the unit circle corresponding to θ. Cosine and sine are between −1 and 1 −1 ≤ cos θ ≤ 1 and − 1 ≤ sin θ ≤ 1 for every angle θ Domain and range of cosine and sine • The domain of both cosine and sine is the set of real numbers. • The range of both cosine and sine is the interval [−1, 1].

What is secant?

The secant of an angle θ, denoted sec θ, is defined by sec θ = 1/cos θ .

What does the function sinus represent?

The sine of θ, denoted sin θ, is the second coordinate of the endpoint of the radius of the unit circle corresponding to θ. The endpoint of the radius of the unit circle corresponding to θ has coordinates (cos θ, sin θ).

What are positive and negative angles?

• Angle measurements for a radius on the unit circle are made from the positive horizontal axis. • Positive angles correspond to moving counterclockwise from the positive horizontal axis. • Negative angles correspond to moving clockwise from the positive horizontal axis. A radius of the unit circle corresponding to θ degrees also corresponds to θ + 360n degrees for every integer n.

What are the trigonometric identities with π2 − θ or (90 - θ)?

cos( π2 − θ) = sin θ sin( π2 − θ) = cos θ tan( π2 − θ) = 1/tan θ

What are the trigonometric identities with θ + π ?

cos(θ + π) = − cos θ sin(θ + π) = − sin θ tan(θ + π) = tan θ

What does the function tangent represent?

The tangent of an angle θ, denoted tan θ, is defined by tan θ = sin θ/cos θ provided that cos θ != 0 tan θ is the slope of the radius of the unit circle corresponding to θ.

What is the equation of a unit circle?

The unit circle in the xy-plane is the set of points (x, y) such that x^2 + y^2 = 1.

What is the unit circle?

The unit circle is the circle of radius 1 centered at the origin.

What is the domain and range of tangent and why?

tangent real numbers that are not odd multiples of π2 (−∞, ∞)