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