Transcript 9-1 PPT Trig and Tangent

Investigate Tangent Ratios
1. Select one angle measure from 20º, 30º, 40º, or 50º .
2. Each person draw a right triangle (∆ABC) where A has the selected
measure.
3. Each person make the triangle different sizes. Then measure the legs
using inches.
4. Compute the ratio leg opposite A
leg adjacent A
5. Compare the ratio with others that selected the same angle measure.
Make a conjecture.
Section 9-1 Trigonometry and the Tangent Ratio
SPI 32F: determine the trigonometric ratio for a right triangle needed to
solve a real-world problem given a diagram
Objectives:
• Use tangent ratios to determine side lengths in triangles
Previously, to find measures in a right triangle, we used:
• Pythagorean Theorem
• Distance Formula
• 30-60-90 or 45-45-90 triangle theorems
Now, we will use Trigonometry (triangle measure).
We will investigate 3 of the 6 trigonometric functions:
• tangent
• sine
• cosine
Trigonometry and the Tangent Ratio
Tangent Ratio:
In a right triangle, the ratio of the length of the leg
opposite P to the length of the leg adjacent to P is
constant, no matter what lengths are chosen for the
sides. This is called the tangent ratio.
Tangent of P = opposite leg
adjacent leg
Write a Tangent Ratio
Write the tangent ratios for T and U.
tan T = opp = UV = 3
adj
TV
4
tan U = opp = TV = 4
adj
UV 3
Apply Tangent to Real-World
Suppose you are snow skiing in Colorado and get lost.
You can see a cliff in the distance which is where the ski
lodge is located. About how far away is the cliff?
Step 1: Point your compass towards the cliff. Take a reading.
Step 2: Turn 90 degrees and walk 50 feet in a straight path.
Step 3: Point your compass towards the cliff. Take a reading.
Suppose in step 3, you find the
measure of angle from where
you stand to the cliff is 86. Now
find the distance to the cliff
using the tangent ratio.
tan 86 = x
50
x = 50 tan 86
= 715
The cliff is about 715 feet away.
Inverse of a Tangent
If you know the tangent ratio of an angle, finding the
unknown angle is called the inverse tangent and is written
as tan-1.
Find the mY.
1. First find the tan ratio.
tan Y = 100 = 2.44
41
2. Take the tan inverse of the ratio.
m Y = tan-1 (2.44)
= 68