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Right Triangle Trigonometry Unit 4 Lesson 1
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Page 1: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Right Triangle Trigonometry

Unit 4 Lesson 1

Page 2: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

RIGHT TRIANGLE TRIGONOMETRY

Students will be able to:

Use the right triangle trigonometry for finding the unknowns in a right angled triangle

Key Vocabulary:

• Right Triangle Trigonometry

• Trigonometric Ratios

• Pythagorean Theorem

Page 3: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Right Triangle Trigonometry

Right angle trigonometry is the trigonometry of a right-angled triangle.

A right-angled triangle is a triangle in which one angle is 90 degrees.

RIGHT TRIANGLE TRIGONOMETRY

Page 4: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Properties of a Right Angled Triangle

• A hypotenuse is the line segment opposite to the right-angle.

• An opposite is the line segment opposite to the angle ϴ.

• An adjacent is the line segment next to the angle ϴ.

RIGHT TRIANGLE TRIGONOMETRY

Page 5: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Trigonometric Ratios

There are total 6 trigonometric ratios for a right angled triangle.

1. Sine

2. Cosine

𝒔𝒊𝒏 𝜽 = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆

𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆

𝒄𝒐𝒔 𝜽 = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕

𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆

RIGHT TRIANGLE TRIGONOMETRY

Page 6: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

3. Tangent

4. Cosecant

𝒕𝒂𝒏 𝜽 = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆

𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕

𝒄𝒐𝒔𝒆𝒄 𝜽 = 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆

𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆=

𝟏

𝒔𝒊𝒏(𝜽)

RIGHT TRIANGLE TRIGONOMETRY

Page 7: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

5. Secant

6. Cotangent

𝒔𝒆𝒄 𝜽 = 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆

𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕=

𝟏

𝒄𝒐𝒔(𝜽)

𝒄𝒐𝒕 𝜽 = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕

𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆=

𝟏

𝒄𝒐𝒕(𝜽)

RIGHT TRIANGLE TRIGONOMETRY

Page 8: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Problem 1: Write the trigonometric ratios for the angle 𝑪 in the triangle shown.

RIGHT TRIANGLE TRIGONOMETRY

Page 9: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Problem 1: Write the trigonometric ratios for the angle 𝑪 in the triangle shown.

𝒔𝒊𝒏 𝑪 =

𝟏𝟎

𝟐𝟔=

𝟓

𝟏𝟑

𝒄𝒐𝒔 𝑪 =𝟐𝟒

𝟐𝟔=

𝟏𝟐

𝟏𝟑

𝒕𝒂𝒏 𝑪 =𝟏𝟎

𝟐𝟒=

𝟓

𝟏𝟐

𝒄𝒐𝒔𝒆𝒄 𝑪 =𝟐𝟔

𝟏𝟎=

𝟏𝟑

𝟓

𝒔𝒆𝒄 𝑪 =𝟐𝟔

𝟐𝟒=

𝟏𝟑

𝟏𝟐

𝒄𝒐𝒕 𝑪 =𝟐𝟒

𝟏𝟎=

𝟏𝟐

𝟓

RIGHT TRIANGLE TRIGONOMETRY

Page 10: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Problem 2: Find the value of 𝒙. Round to the nearest tenth.

RIGHT TRIANGLE TRIGONOMETRY

Page 11: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Problem 2: Find the value of 𝒙. Round to the nearest tenth.

𝒔𝒊𝒏𝜽 = 𝒐𝒑𝒑

𝒉𝒚𝒑

𝒔𝒊𝒏𝟏𝟕° =𝒙

𝟏𝟏

𝒙 = 𝟏𝟏 × 𝒔𝒊𝒏𝟏𝟕° 𝒙 = 𝟑. 𝟐

RIGHT TRIANGLE TRIGONOMETRY

Page 12: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Pythagorean Theorem

In a right-triangle, the sum of the squares of the lengths of adjacent and opposite is equal to the square of the length of hypotenuse.

Where,

𝒄 = 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆

𝒂 = 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆

𝒃 = 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐

RIGHT TRIANGLE TRIGONOMETRY

Page 13: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Problem 3: Find the unknown length 𝒙 in the right triangle shown.

RIGHT TRIANGLE TRIGONOMETRY

Page 14: Right Triangle Trigonometry - PreCalculusCoach.com...Right Triangle Trigonometry Right angle trigonometry is the trigonometry of a right-angled triangle. A right-angled triangle is

Problem 3: Find the unknown length 𝒙 in the right triangle shown.

By Pythagorean theorem,

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 𝟏𝟑𝟐 = 𝒙𝟐 + 𝟏𝟐𝟐 𝒙𝟐 = 𝟏𝟔𝟗 − 𝟏𝟒𝟒

𝒙𝟐 = 𝟐𝟓 𝒙 = 𝟓

RIGHT TRIANGLE TRIGONOMETRY


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