Download Section 10.5 Trigonometric Ratios via Similarity. 1. An

Section 10.5 Trigonometric Ratios via Similarity.
1. An example leading to Trigonometric Ratios.
Suppose a group of students wanted to find the angle of elevation of the sun at a certain time of the
day as well as the length of a vertical flagpole. This is what they decided to do.
• They set up a stick of length 2 m vertically, and measured its shadow, getting 1.5 m.
• They also measured the shadow of the flagpole and discovered it to be 4.5 m.
• Then, using 1 cm representing 1 m, they made a scale drawing of the stick and of the shadow (see
4XY Z below) to find the angle of elevation of the sun (the angle 6 XY Z in the left triangle).
They got 53◦ .
• With the same scale and the same angle of elevation as in 3), they made a scale drawing to find
the length of the flagpole (see 4ABC below)
2. Analyzing the example leading to Trigonometric Ratios.
Now let us study their drawings: 4XY Z and 4ABC (in my figure they are not drawn to scale). It
is given that 6 XY Z = 6 ACB (the angle of elevation of the sun). Answer the following questions.
• Are the triangles similar? Give the reason.
• Are the following true:
AB
BC
YZ
BC
=
(reason?),
=
(reason?)
YZ
XY
AB
XY
YZ
BC
=
? Find this length.
AB
XY
• If the shadow of the flagpole was not measured at the same time as the shadow of the stick,
would you still be able to calculate the length of the flagpole this way?
length of pole
• Do you agree that, for any vertical pole, the ratio
depends on the angle of
length of shadow
elevation of the sun?
• Can you find the length of the flagpole using the proportion
We come to the conclusion that if the angle changes, the ratio changes too, or in other words,
this ratio is the function of the angle of elevation. Since any two right triangles with the same
acute angle are similar (AA test), the ratio of two sides of any of these similar triangles does not
depend on the size of a triangle, it only depends on the angle.
3. Trigonometric Ratios.
In any right triangle ABC with a right angle at C, we refer to b = AC as the side adjacent to 6 A,
a = BC as the side opposite to 6 A and c = AB as the hypotenuse.
Each of the ratios
side opposite 6 A side adjacent 6 A side opposite 6 A
,
,
hypotenuse
hypotenuse
side adjacent 6 A
does not depend on the size of the triangle, it depends only on the size of 6 A.
Definition of Trigonometric Ratios.
In a right triangle ABC with 6 C = 90◦ , we define
side opposite 6 A
hypotenuse
side adjacent 6 A
• cos(6 A) =
hypotenuse
side opposite 6 A
• tan(6 A) =
side adjacent 6 A
• sin(6 A) =
(sine of 6 A)
(cosine of 6 A)
(tangent of 6 A)
These three ratios, sine, cosine, and tangent of a given angle, are called trigonometric ratios. Trigonometry is the study of these and other functions arising from relationships between sides and angles of
triangles.
To compute a trigonometric ratio for an angle, you may use a protractor and construct ANY right
triangle with such angle, then measure the sides, and compute the ratio. Before the era of calculators
and computers, people used trigonometric tables, now we can easily find sine, cosine, and tangent of
an angle using calculators or computers.
4. Exercises.
sin6 A
cos6 A
2. Use Pythagorean Theorem a2 + b2 = c2 (here a, b are the legs and c is the hypotenuse of a right
triangle) to show that sin2 6 A + cos2 6 A = 1.
1. Show that tan6 A =
3. Study examples 10.15-10.18.
4. Be able to find the values of sine, cosine and tangent of 30, 60 and 45 degree angles using
appropriate right triangles.
5. Do the homework.