0.1 Trigonometry: The Sine Function
Objectives: Calculate the sine ratio of a given angle and use it to solve a triangle.
Use the patterns of special right triangles to calculate sine ratios.
The branch of mathematic
...
0.1 Trigonometry: The Sine Function
Objectives: Calculate the sine ratio of a given angle and use it to solve a triangle.
Use the patterns of special right triangles to calculate sine ratios.
The branch of mathematics called trigonometry consists of three main functions, which
are known as the sine, cosine, and tangent functions. In a right triangle, the sine of an
acute angle is defined as the ratio of the opposite leg divided by the hypotenuse. The
Greek letter “theta,” denoted θ, is typically used to represent the acute angle. The
abbreviation for sine is sin, so the expression sin θ is read as “the sine of theta.”
sin θ =
opposite
hypotenuse
Read: “The sine of theta equals opposite divided by hypotenuse.”
hypotenuse opposite leg
θ
adjacent leg
EXAMPLE 1: Calculate sin θ and then use it to find x in the second triangle.
40 x
5 4
θ θ
3
In the first triangle, the side opposite θ is 4, while the hypotenuse is 5. We would write
4
sin
5
opp
hyp
= =
Now we can show that the two triangles are similar triangles by using a theorem from
Geometry called Angle-Angle (AA) Similarity. Each triangle has an angle θ, and these
angles must be congruent. Additionally, the two right angles are congruent. Therefore,
the two triangles share two pairs of congruent angles, which gives AA Similarity.
Once we know that the two triangles are similar, the definition of similarity states that the
corresponding sides are proportional. As a result, we can use the sine ratio to write and
solve a proportion.
549 4
5 40
x
=
(Use the similar triangles to write a proportion.)
5x = 160 (Cross-multiply.)
x = 32 (Divide both sides by 5.)
Notice that the first triangle is a 3-4-5 triangle. Right triangles with integer lengths for all
three sides form what we call Pythagorean Triples. Memorizing the Pythagorean
Triples is useful because it reduces the need to use the Pythagorean Theorem.
COMMON PYTHAGOREAN TRIPLES
3-4-5 6-8-10 9-12-15 12-16-20 15-20-25 etc.*
5-12-13
8-15-17
7-24-25
*The 3-4-5 triple can be used to derive infinitely many other triples by multiplying all
sides by any integer. As long as all three sides are multiplied by the same value, all three
sides will remain in proportion, thereby producing yet another triple. This process gives
additional common triples such as 6-8-10, 9-12-15, etc.
(The same process can be used to find multiples of the 5-12-13, 8-15-17, and 7-24-25
triples, but we do not commonly use these triples.)
In addition to the Pythagorean Triples, we want to recall and be able to use special right
triangles such as the 30-60-90 triangle and the 45-45-90 triangle.
THEOREM: 45°-45°-90° TRIANGLES
In any 45°-45°-90° triangle, the hypotenuse is
2
times each leg.
45°
a a
2
45°
a
We could paraphrase the theorem with the following two rules:
1. In any 45°-45°-90° triangle, given the legs, multiply by
2
to find the hypotenuse.
2. In any 45°-45°-90° triangle, given the hypotenuse, divide by
2
to find each le
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