Transcript 9.5 Trigonometric Ratios

9.5 Trigonometric
Ratios
Unit IIC Day 4
Do Now
 Are all 30-60-90 triangles congruent?
 Are all 30-60-90 triangles similar?
 Explain.
Trigonometric Ratios
 A trigonometric ratio is a ratio of the lengths of two
sides of a right triangle.
 The three basic trigonometric ratios are called sine,
cosine, and tangent.
Trigonometric Ratios
 Let ∆ABC be a right triangle.
 The sine, the cosine, and the tangent of the acute
angle θare defined as follows:
opposite
sin(q ) =
hypotenuse
adjacent
cos(q ) =
hypotenuse
opposite
tan(q ) =
adjacent
Ex. 1: Finding Trig Ratios
 Find the sine, the cosine, and the tangent of 45°
without using a calculator.
 Do we have enough information to draw a picture?
Ex. 2: Finding Trig Ratios
 Find the sine, the cosine, and the tangent of 60°
without using a calculator.
 Start by drawing a picture.
Ex. 3: Finding Trig Ratios
 Find the sine, cosine, and tangent of 30° without using
a calculator.
 Start by drawing a picture.
Ex. 4: Using Trig. Ratios
 Find the value of x. Round decimals to the nearest
tenth.
 We know the value of the _____________ side and need
the value of the ____________ side. Which ratio is that?
Ex. 4A: Using Trig. Ratios
 Find the value of y. Round decimals to the nearest
tenth.
 We know the value of the _____________ side and need
the value of the ____________ side. Which ratio is that?
Notes:
 If you look back at the previous examples, you will
notice that the sine or the cosine of an acute angle is
always less than 1. Why is this?
Trig. Ratios in Real Life
 Suppose look up at a point in the distance, such as the
top of a tree.
 The angle that your line of sight makes with a
horizontal line is called angle of elevation.
Ex. 5: Indirect Measurement
 You lie on the ground 45 feet
from the base of the tree. You
measure the angle of elevation
from a point on the ground to the
top of the top of the tree to be
59°. Estimate the height of the
tree.
Ex. 6: Estimating Distance
 The escalator at the Wilshire/Vermont Metro Rail
Station in Los Angeles rises 76 feet at a 30° angle.
Find the distance d a person travels on the escalator
stairs.
Closure
 Explain how the calculator “knows” the values of
trigonometric ratios (e.g., sin(74°)) if it doesn’t have a
picture of a triangle.