Download Analysis Functions of Acute Angles www.AssignmentPoint.com The

Analysis Functions of Acute Angles
www.AssignmentPoint.com
www.AssignmentPoint.com
The characteristics of similar triangles, originally formulated by Euclid, are the building
blocks of trigonometry. Euclid's theorems state if two angles of one triangle have the
same measure as two angles of another triangle, then the two triangles are similar. Also,
in similar triangles, angle measure and ratios of corresponding sides are preserved.
Because all right triangles contain a 90° angle, all right triangles that contain another
angle of equal measure must be similar. Therefore, the ratio of the corresponding sides of
these triangles must be equal in value. These relationships lead to the trigonometric
ratios. Lowercase Greek letters are usually used to name angle measures. It doesn't
matter which letter is used, but two that are used quite often are alpha (α) and theta (θ).
Angles can be measured in one of two units: degrees or radians. The relationship
between these two measures may be expressed as follows:
The following ratios are defined using a circle with the equation x 2 + y 2 = r 2 and refer to
Figure
1.
www.AssignmentPoint.com
Figure
Reference
1
triangles.
Remember, if the angles of a triangle remain the same, but the sides increase or decrease
in length proportionally, these ratios remain the same. Therefore, trigonometric ratios in
www.AssignmentPoint.com
right triangles are dependent only on the size of the angles, not on the lengths of the
sides.
The cosecant,
reciprocals
secant,
of
and cotangent are trigonometric
the sine,
cosine,
functions that
and tangent,
are
the
respectively.
If trigonometric functions of an angle θ are combined in an equation and the equation is
valid for all values of θ, then the equation is known as a trigonometric identity. Using
the trigonometric ratios shown in the preceding equation, the following trigonometric
identities
can
be
constructed.
Symbolically, (sin α) 2 and sin 2 α can be used interchangeably. From Figure (a) and the
Pythagorean
theorem,
x2+
www.AssignmentPoint.com
y2=
r 2.
These
three
trigonometric
identities
are
extremely
important:
Example 1: Find sin θ and tan θ if θ is an acute angle (0° ≤ θ ≤ 90°) and cos θ = ¼.
www.AssignmentPoint.com
Example 2: Find sin θ and cos θ if θ is an acute angle (0° ≤ θ ≤ 90°) tan θ = 6.
If the tangent of an angle is 6, then the ratio of the side opposite the angle and the side
adjacent to the angle is 6. Because all right triangles with this ratio are similar, the
hypotenuse can be found by choosing 1 and 6 as the values of the two legs of the right
triangle
and
then
applying
the
www.AssignmentPoint.com
Pythagorean
theorem.
Trigonometric functions come in three pairs that are referred to as cofunctions. The sine
and cosine are cofunctions. The tangent and cotangent are cofunctions. The secant and
cosecant are cofunctions. From right triangle XYZ, the following identities can be
derived:
Using
Figure
2,
observe
that
∠X
and
www.AssignmentPoint.com
∠Y
are
complementary.
Figure
2
Reference
Thus,
in
triangles.
general:
Example 3: What are the values of the six trigonometric functions for angles that
measure 30°, 45°, and 60° (see Figure 3 and Table 1 ).
www.AssignmentPoint.com
TABLE 1 Trigonometric Ratios for 30°, 45°, and 60° Angles
Figure
Drawings for Example 3
www.AssignmentPoint.com
3