trigonometric ratios
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Ihr Logo
TrigonometricRatios
Your Logo Topic 004 Page 2
Outline:
The Three Trigonometric Ratios
Reciprocal and Complementary Ratios
Trigonometric Ratios of Special Angles
– Angle
– Angle
– Angle
Some Exercises
Your Logo Topic 004 Page 3
The Three Trigonometric Ratios
𝜽𝑪 ′
𝑩 ′
𝑨 ′
𝒄 ′ 𝒂 ′
𝒃 ′
𝜽𝑨 𝑪
𝑩
𝒃
𝒂𝒄
Your Logo Topic 004 Page 4
Since the two angles are equal (the right angle and ), then the third angles are equal and the two triangles are similar. Thus the corresponding sides of the two triangles are proportional, namely
𝒂𝒄
=𝒂 ′𝒄 ′;𝒃𝒄
=𝒃 ′𝒄 ′;𝒂𝒃
=𝒂 ′𝒃′
Your Logo Topic 004 Page 5
sine of or sin
cosine of or cos
tangent of or tan
Your Logo Topic 004 Page 6
Example 4.1:
𝜽𝑪
𝑩
𝑨
𝒄=? 𝒂=𝟑
𝒃=𝟒
Given , determine the following:
a. b. cos c. tan
Your Logo Topic 004 Page 7
Example 4.2:
𝑳
𝑴
𝑵
𝟏𝟑𝟓
?
Determine the three basic trigonometric ratios for the two angles M and N.
Your Logo Topic 004 Page 8
Reciprocal and Complementary Angles
𝜽𝑨 𝑪
𝑩
𝒃
𝒂𝒄
Your Logo Topic 004 Page 9
secant of or sec
cosecant of or csc
cotangent of or cot
Your Logo Topic 004 Page 10
Example 4.3:
𝜽𝑪
𝑩
𝑨
𝒄=? 𝒂=𝟑
𝒃=𝟒
Given , find the reciprocal ratios of angle
a. b. csc c. cot
Your Logo Topic 004 Page 11
Example 4.4:
𝑳
𝑴
𝑵
𝟏𝟑𝟓
?
Determine the three reciprocal ratios for the two angles M and N.
Your Logo Topic 004 Page 12
Do you notice anything with the definitions of secant, cosecant, and cotangent? How are they related to the three basic trigonometric ratios?
Your Logo Topic 004 Page 13
Thus, cosecant is the reciprocal ratio of sine; secant is the reciprocal ratio of cosine; and cotangent is the reciprocal ratio of tangent.Then, cosecant, secant, and cotangent are called reciprocal ratios.
Your Logo Topic 004 Page 14
Ratios of Complementary Angles
𝜽𝑨 𝑪
𝑩
𝒃
𝒂𝒄
𝜶Consider the given
right triangle. This time focus not only on but also on . Based on the figure, how are these two angles related?
Your Logo Topic 004 Page 15
You have learned that . If you consider the complementary angle , then . Therefore the sine of an acute angle is equal to the cosine of its complementary angle. In short, if is a complement of , or , then
Similarly,
and
Your Logo Topic 004 Page 16
Example 4.5:
Given , determine the six trigonometric ratios of the two acute angles and .
𝑪
𝑩
𝑨
𝟓𝟒
𝟑
Your Logo Topic 004 Page 17
Trigonometric Ratios of the Special Angles
a. The 𝑩
𝑨 𝑪
𝑩
𝟐 𝒙
𝟐 𝒙 𝟐 𝒙
𝟔𝟎° 𝟔𝟎°
𝟔𝟎°Consider the equilateral triangle with sides . The measures of the angles, , , and , are each .
Your Logo Topic 004 Page 18
𝑩
𝑨 𝑪𝑫
𝑩
𝟐 𝒙
𝟐 𝒙 𝟐 𝒙
𝟔𝟎° 𝟔𝟎°
Suppose line segment BD is a perpendicular bisector. It divides and side AC into two equal parts with . Hence and .
Your Logo Topic 004 Page 19
Applying the Pythagorean theorem,
𝟔𝟎°
𝟐 𝒙
𝒙𝑨 𝑫
𝑩
Your Logo Topic 004 Page 20
Since all sides are now known, the trigonometric ratios of the can be obtained.
Your Logo Topic 004 Page 21
The reciprocal ratios are
Your Logo Topic 004 Page 22
Consider again . The other acute angle, is since it is the complement of . Hence the trigonometric ratios of the are as follows:
𝟐 𝒙
𝒙𝑨 𝑫
𝑩
𝒙 √𝟑
b. The 3
Your Logo Topic 004 Page 23
The reciprocal ratios are
Your Logo Topic 004 Page 24
This time consider the isosceles right triangle with equal sides . The two acute angle are likewise equal and each measures .
c. The
Applying the Pythagorean theorem,
𝒙𝑨 𝑪
𝑩
𝟒𝟓°
𝟒𝟓°
Your Logo Topic 004 Page 25
The trigonometric ratios of the are
Your Logo Topic 004 Page 26
The reciprocal ratios are
Your Logo Topic 004 Page 27
Trigonometric Ratios of Special Angles
Your Logo Topic 004 Page 28
Example 4.6:
Given right , with and . Find sides AC and AB. Determine also the six trigonometric ratios.
𝑨 𝑪
𝑩
𝟓𝒄𝒎
𝟒𝟓°
Your Logo Topic 004 Page 29
Example 4.7:
Solve for x and y in the given triangle. Determine the six trigonometric ratios.
𝒙
𝟑𝟎°𝟐𝟒𝒄𝒎
𝒚
Your Logo Topic 004 Page 30
Thank You!!!Courage is the first of human virtues
because it makes all others possible.
- Aristotle
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