6.4 – Trigonometric Functions Now, we will consider an acute angle in standard position. And choose a point P with coordinates x , y on the terminal side, and draw a right triangle. The side adjacent to has length x and the side opposite has length y. The length of the hypotenuse, r, is the distance from the origin, which can be found by

x2 y2 r 2 , so r x2 y2 .

y hypotenuse opposite

Now, the trigonometric ratios can be written in terms of x, y, and r. y x y sin cos tan r r x x r r cot sec csc y x y

x adjacent

EXAMPLE 1: Find the sine, cosine and tangent of the angle , whose terminal side passes through 2, 1 . (HINT: First, you need to find r.)

Let t be a real number such that it is an angle of t in radians in standard position. If any point x , y is on y r r csc t y sint

the terminal side of angle t, then:

x r r sec t x

cos t

y x x cot t y

tant

where r x2 y2 is the distance from x , y to the origin.

x2 y2 1 , so now sint

In the UNIT CIRCLE, the radius

y y x x y and cos t x . r 1 r 1

Now, we have new Unit Circle descriptions of Trigonometric Functions: P has coordinates cos t , sin t tan t

y sin t x cos t

cot t

x cos t y sin t

sec t

1 1 x cos t

csc t

1 1 y sin t

NOW we can talk about domain and range of these functions: The domain of the sine and cosine functions is the set of all real numbers The range of the sine and cosine functions is the set of all real numbers between – 1 and 1 (inclusive), that is, the interval 1, 1

Think about it, since the sine and cosine ratios have the radius of the circle in the denominator, the numerator will never be greater than the denominator!!! y The tangent function is defined as tan t , whenever x 0 , that is, for all points on the unit x 3 circle except at 0, 1 and 0, 1 , which is at and all angles coterminal to these. and 2 2 The domain of the tangent function is the set of all real numbers except:

2

2k , where k 0, 1, 2, 3,...

The range of the tangent function is the set of all real numbers. The next thing you need to pay attention to is the sign of each function in each quadrant, and of course, this is related to the sign of x and y in each quadrant: y Quadrantal Angles

Quad II

t

,

2 sint cos t tant

Quad III

3 t 2 sint cos t tant

,

0t

Quad I

,

2 sint cos t tant

3 t 2 2 sint cos t tant

x

Quad IV

,

EXAMPLE 2: Find the exact value of the trig functions at the quadrantal angles on the Unit Circle ( radius 1 ) T sin t

cos t tan t cot t

sec t csc t

0

2

3 2

2

For an angle in standard position, the reference angle t ' or ' is the positive acute angle formed by the terminal side of t or and the x-axis.

t' t

(NOTE: These angles are NEVER made with the y-axis!!!!!)

t

t

t

t’ t’

t’

t' t

t' t

t' t

t ' 2 t

'

' 180

' 180

' 360

To find the sine, cosine or tangent of t radians: o Sketch the angle t in standard position and determine in which quadrant the terminal side falls. o Find the reference angle, t’. o Find the sine, cosine, or tangent of t’ and give it the appropriate sign. EXAMPLE 3: Use reference angles to find the exact values of sin t , cos t , and tan t . 5 5 5 a. t b. t c. t 4 3 6

If an angle is larger than 2 , it is possible to find a coterminal angle between 0 and 2 , and then use its reference angle to find the trig values. EXAMPLE 4: Find the sine, cosine and tangent of

9 4

The COMPLETE Unit Circle

30 60 90

45 45 90

30 60 90