The Six Basic Trigonometric Functions

Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle.

Evaluate limit lim θ→π/4 θtan(θ) Since θ = π/4 is in the domain of the function θtan(θ) we use Substitution Theorem to substitute π/4 for θ in the limit expression: lim θ→π/4 θtanθ = π 4 tan π 4 = π 4 1 = π 4. – Typeset by FoilTEX – 10. How do you find the limit of #( 1 - (5/x) ) ^x# as x approaches infinity? Calculus Limits Determining Limits Algebraically. The limit is needed for finding the derivative of $ sin x$, so if you type 'derivative' and 'sin' into your favorite search engine, lots of webpages with evaluations of the limit will turn up.

This calculus video tutorial provides a basic introduction into evaluating limits of trigonometric functions such as sin, cos, and tan. It contains plenty o. The AP Calculus Problem Book Publication history: First edition, 2002 Second edition, 2003 Third edition, 2004 Third edition Revised and Corrected, 2005.

To define the trigonometric functions, first consider the unit circle centered at the origin and a point (P=(x,y)) on the unit circle. Let (θ) be an angle with an initial side that lies along the positive (x)-axis and with a terminal side that is the line segment (OP). An angle in this position is said to be in standard position (Figure (PageIndex{2})). We can then define the values of the six trigonometric functions for (θ) in terms of the coordinates (x) and (y).

Definition: Trigonometric functions

Let (P=(x,y)) be a point on the unit circle centered at the origin (O). Let (θ) be an angle with an initial side along the positive (x)-axis and a terminal side given by the line segment (OP). The trigonometric functions are then defined as

(sin θ=y)	(cscθ=dfrac{1}{y})
(cos θ=x)	(secθ=dfrac{1}{x})
(tan θ=dfrac{y}{x})	(cot θ=dfrac{x}{y})

If (x=0, secθ) and (tan θ) are undefined. If (y=0), then (cot θ) and (cscθ) are undefined.

We can see that for a point (P=(x,y)) on a circle of radius (r) with a corresponding angle (θ), the coordinates (x) and (y) satisfy

[begin{align} cos θ &=dfrac{x}{r} x&=rcos θ end{align}]

and

[begin{align} sin θ&=dfrac{y}{r} y&=rsin θ. end{align}]

Figure (PageIndex{3.1}): Diagram demonstrating trigonometric functions in the unit circle., ).

The values of the other trigonometric functions can be expressed in terms of (x,y), and (r) (Figure (PageIndex{3})).

Figure (PageIndex{3.2}): For a point (P=(x,y)) on a circle of radius (r), the coordinates (x) and y satisfy (x=rcos θ) and (y=rsin θ).

Table shows the values of sine and cosine at the major angles in the first quadrant. From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. The values of the other trigonometric functions are calculated easily from the values of (sin θ) and (cos θ.)

**Table (PageIndex{2})**: Values of (sin θ) and (cos θ) at Major Angles (θ) in the First Quadrant
(θ)θ	(sin θ)	(cos θ)
0	0	1
(dfrac{π}{6})	(dfrac{1}{2})	(dfrac{sqrt{3}}{2})
(dfrac{π}{4})	(dfrac{sqrt{2}}{2})	(dfrac{sqrt{2}}{2})
(dfrac{π}{3})	(dfrac{sqrt{3}}{2})	(dfrac{1}{2})
(dfrac{π}{2})	1	0

Example (PageIndex{2}): Evaluating Trigonometric Functions

Evaluate each of the following expressions.

(sin(dfrac{2π}{3}))
(cos(−dfrac{5π}{6}))
(tan(dfrac{15π}{4}))

Solution:

a) On the unit circle, the angle (θ=dfrac{2π}{3}) corresponds to the point ((−dfrac{1}{2},dfrac{sqrt{3}}{2})). Therefore, (sin)((dfrac{2π}{3}))=(y)=(dfrac{sqrt{3}}{2}).

b) An angle (θ=−dfrac{5π}{6}) corresponds to a revolution in the negative direction, as shown. Therefore, (cos(−dfrac{5π}{6}))=(x)=(−dfrac{sqrt{3}}{2}).

c) An angle (θ)=(dfrac{15π}{4})=(2π)+(dfrac{7π}{4}). Therefore, this angle corresponds to more than one revolution, as shown. Knowing the fact that an angle of (dfrac{7π}{4}) corresponds to the point ((dfrac{sqrt{2}}{2},dfrac{sqrt{2}}{2})), we can conclude that (tan(dfrac{15π}{4})=dfrac{y}{x}=−1).

Exercise (PageIndex{2})

Evaluate (cos(3π/4)) and (sin(−π/6)).

Hint

Look at angles on the unit circle.

Answer

The Six Basic Trigonometric Functions

Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle.

To define the trigonometric functions, first consider the unit circle centered at the origin and a point (P=(x,y)) on the unit circle. Let (θ) be an angle with an initial side that lies along the positive (x)-axis and with a terminal side that is the line segment (OP). An angle in this position is said to be in standard position (Figure (PageIndex{2})). We can then define the values of the six trigonometric functions for (θ) in terms of the coordinates (x) and (y).

Definition: Trigonometric functions

Let (P=(x,y)) be a point on the unit circle centered at the origin (O). Let (θ) be an angle with an initial side along the positive (x)-axis and a terminal side given by the line segment (OP). The trigonometric functions are then defined as

(sin θ=y)	(cscθ=dfrac{1}{y})
(cos θ=x)	(secθ=dfrac{1}{x})
(tan θ=dfrac{y}{x})	(cot θ=dfrac{x}{y})

If (x=0, secθ) and (tan θ) are undefined. If (y=0), then (cot θ) and (cscθ) are undefined.

We can see that for a point (P=(x,y)) on a circle of radius (r) with a corresponding angle (θ), the coordinates (x) and (y) satisfy

[begin{align} cos θ &=dfrac{x}{r} x&=rcos θ end{align}]

and

[begin{align} sin θ&=dfrac{y}{r} y&=rsin θ. Get faster support with live chat video chat. end{align}]

Figure (PageIndex{3.1}): Diagram demonstrating trigonometric functions in the unit circle., ).

The values of the other trigonometric functions can be expressed in terms of (x,y), and (r) (Figure (PageIndex{3})).

Figure (PageIndex{3.2}): For a point (P=(x,y)) on a circle of radius (r), the coordinates (x) and y satisfy (x=rcos θ) and (y=rsin θ).

Table shows the values of sine and cosine at the major angles in the first quadrant. From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. The values of the other trigonometric functions are calculated easily from the values of (sin θ) and (cos θ.)

**Table (PageIndex{2})**: Values of (sin θ) and (cos θ) at Major Angles (θ) in the First Quadrant
(θ)θ	(sin θ)	(cos θ)
0	0	1
(dfrac{π}{6})	(dfrac{1}{2})	(dfrac{sqrt{3}}{2})
(dfrac{π}{4})	(dfrac{sqrt{2}}{2})	(dfrac{sqrt{2}}{2})
(dfrac{π}{3})	(dfrac{sqrt{3}}{2})	(dfrac{1}{2})
(dfrac{π}{2})	1	0

Example (PageIndex{2}): Evaluating Trigonometric Functions

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