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TrigonometricRatios

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Outline:

The Three Trigonometric Ratios

Reciprocal and Complementary Ratios

Trigonometric Ratios of Special Angles

– Angle

– Angle

– Angle

Some Exercises

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The Three Trigonometric Ratios

𝜽𝑪 ′

𝑩 ′

𝑨 ′

𝒄 ′ 𝒂 ′

𝒃 ′

𝜽𝑨 𝑪

𝑩

𝒃

𝒂𝒄

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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′

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sine of or sin

cosine of or cos

tangent of or tan

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Example 4.1:

𝜽𝑪

𝑩

𝑨

𝒄=? 𝒂=𝟑

𝒃=𝟒

Given , determine the following:

a. b. cos c. tan

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Example 4.2:

𝑳

𝑴

𝑵

𝟏𝟑𝟓

?

Determine the three basic trigonometric ratios for the two angles M and N.

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Reciprocal and Complementary Angles

𝜽𝑨 𝑪

𝑩

𝒃

𝒂𝒄

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secant of or sec

cosecant of or csc

cotangent of or cot

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Example 4.3:

𝜽𝑪

𝑩

𝑨

𝒄=? 𝒂=𝟑

𝒃=𝟒

Given , find the reciprocal ratios of angle

a. b. csc c. cot

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Example 4.4:

𝑳

𝑴

𝑵

𝟏𝟑𝟓

?

Determine the three reciprocal ratios for the two angles M and N.

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Do you notice anything with the definitions of secant, cosecant, and cotangent? How are they related to the three basic trigonometric ratios?

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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.

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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?

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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

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Example 4.5:

Given , determine the six trigonometric ratios of the two acute angles and .

𝑪

𝑩

𝑨

𝟓𝟒

𝟑

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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 .

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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 .

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Applying the Pythagorean theorem,

𝟔𝟎°

𝟐 𝒙

𝒙𝑨 𝑫

𝑩

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Since all sides are now known, the trigonometric ratios of the can be obtained.

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The reciprocal ratios are

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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

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The reciprocal ratios are

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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°

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The trigonometric ratios of the are

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The reciprocal ratios are

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Trigonometric Ratios of Special Angles

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Example 4.6:

Given right , with and . Find sides AC and AB. Determine also the six trigonometric ratios.

𝑨 𝑪

𝑩

𝟓𝒄𝒎

𝟒𝟓°

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Example 4.7:

Solve for x and y in the given triangle. Determine the six trigonometric ratios.

𝒙

𝟑𝟎°𝟐𝟒𝒄𝒎

𝒚

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Thank You!!!Courage is the first of human virtues

because it makes all others possible.

- Aristotle