# Trigonometric Ratios Trigonometric Ratios How do we use trig ratios? M2 Unit 2: Day 3 In a Triangle, we know that the angles have a sum of 180 and that the two acute angles are complementary q So, if one angle is q Then the other one is

90 - q 90 - q q 90 - q Assume mX = q X then mY = 90 - q what if: mX = 90 - q then mX = q Z

Y Trigonometric Ratios: Are ratios of the lengths of 2 sides of a right triangle. There are 3 basic trig ratios: sine, cosine, and tangent (abbreviated sin, cos, and tan) The value of a trig ratio depends only on the measure of the acute angle, not on the particular triangle being used to compute the value. SOHCAHTOA

If you can remember his name, then you can remember your trig ratios! Opposite of A sin A Hypotenuse Adjacent to A cos A Hypotenuse Opposite of A tan A Adjacent to A

Opposite means across from the angle Adjacent means attached to the angle Hypotenuse is always opposite the right angle. Label the hypotenuse, opposite and adjacent for angle A. B C

A Label the hypotenuse, opposite and adjacent for angle q. q Label the hypotenuse, opposite and adjacent for angle X. X Z Y

Now, label the hypotenuse, opposite and adjacent for angle y. opp 15 sin P 0.8824 hyp 17 adj 8 cos P 0.4706 hyp 17 opp 15 tan P 1.875 adj 8 Now find the sine, the cosine, and the tangent

ofQ opp 8 sin Q 0.4706 hyp 17 adj 15 cos Q 0.8824 hyp 17 opp 8 tan Q 0.5333 adj 15 Notice

something about the sine and cosine ratios? How about the tangent ratios? . 2 13 5 opp 12 sin P

0.9231 hyp 13 adj 5 cos P 0.3846 hyp 13 opp 12 tan P 2.4 adj 5 12 Now find the sine, the cosine, and the tangent ofQ

opp 5 sin Q 0.3846 hyp 13 adj 12 cos Q 0.9231 hyp 13 opp 5 tan Q 0.4167 adj 12 Notice something about the sine and

cosine ratios? How about the tangent ratios? . Find tan q . Round to four decimal places. C 42 A 40 q 58

40 Tan q = 0.9524 42 B Find sin 90 - q and tan q . Write each answer as a decimal rounded to four decimal places. B 45 C

q 53 sin ( 90 - q) .8491 tan ( q) .6222 28 A You can use your calculator to find a decimal approximation for trig ratios. Example 3: Use your calculator to approximate the given value to four decimal places. a) sin 82

Solutions: 1.7321 0.9903 b) cos 30 0.8660 c) tan 60 In summary, notice 4 things: 1.

2. 3. 4. The 2 acute angles of a right triangle are always complementary The sin, cos, and tan of congruent angles in similar triangles are always equal no matter the side lengths The sin and cos ratios of 2 complementary angles are always switched The tan ratios of 2 complementary

angles are always reciprocals of one another sin q = cos ( 90 - q) 2 if sin q= 3 5 if sin A = 2 2 then cos(90 - q) = 3 5 then cos B =

2 cos q = sin ( 90 - q) 4 if cos q= 5 4 then sin(90 - q) = 5 6 if cos B = 7 6

then sin(90 - q) = 7 1 tan q = tan ( 90 - q) 2 if tan q= 3 3 then tan(90 - q) = 2 8

if tan q= 5 5 then tan(90 - q) = 8 Homework: Page 159 (#1, 3, 7) and Page 166 (#2-14 even)