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Right Triangle Trigonometry Unit 4 Lesson 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

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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:

β€’ Right Triangle Trigonometry

β€’ Trigonometric Ratios

β€’ 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

β€’ A hypotenuse is the line segment opposite to the right-angle.

β€’ An opposite is the line segment opposite to the angle Ο΄.

β€’ An adjacent is the line segment next to the angle Ο΄.

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

π’”π’Šπ’ 𝜽 = π’π’‘π’‘π’π’”π’Šπ’•π’†

π’‰π’šπ’‘π’π’•π’†π’π’–π’”π’†

𝒄𝒐𝒔 𝜽 = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕

π’‰π’šπ’‘π’π’•π’†π’π’–π’”π’†

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 = π’π’‘π’‘π’π’”π’Šπ’•π’†

𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕

𝒄𝒐𝒔𝒆𝒄 𝜽 = π’‰π’šπ’‘π’π’•π’†π’π’–π’”π’†

π’π’‘π’‘π’π’”π’Šπ’•π’†=

𝟏

π’”π’Šπ’(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 = π’‰π’šπ’‘π’π’•π’†π’π’–π’”π’†

𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕=

𝟏

𝒄𝒐𝒔(𝜽)

𝒄𝒐𝒕 𝜽 = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕

π’π’‘π’‘π’π’”π’Šπ’•π’†=

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 π‘ͺ 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 π‘ͺ in the triangle shown.

π’”π’Šπ’ π‘ͺ =

𝟏𝟎

πŸπŸ”=

πŸ“

πŸπŸ‘

𝒄𝒐𝒔 π‘ͺ =πŸπŸ’

πŸπŸ”=

𝟏𝟐

πŸπŸ‘

𝒕𝒂𝒏 π‘ͺ =𝟏𝟎

πŸπŸ’=

πŸ“

𝟏𝟐

𝒄𝒐𝒔𝒆𝒄 π‘ͺ =πŸπŸ”

𝟏𝟎=

πŸπŸ‘

πŸ“

𝒔𝒆𝒄 π‘ͺ =πŸπŸ”

πŸπŸ’=

πŸπŸ‘

𝟏𝟐

𝒄𝒐𝒕 π‘ͺ =πŸπŸ’

𝟏𝟎=

𝟏𝟐

πŸ“

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.

π’”π’Šπ’πœ½ = 𝒐𝒑𝒑

π’‰π’šπ’‘

π’”π’Šπ’πŸπŸ•Β° =𝒙

𝟏𝟏

𝒙 = 𝟏𝟏 Γ— π’”π’Šπ’πŸπŸ•Β° 𝒙 = πŸ‘. 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 = π‘―π’šπ’‘π’π’•π’†π’π’–π’”π’†

𝒂 = π‘Άπ’‘π’‘π’π’”π’Šπ’•π’†

𝒃 = 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕

π’„πŸ = π’‚πŸ + π’ƒπŸ

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,

π’„πŸ = π’‚πŸ + π’ƒπŸ πŸπŸ‘πŸ = π’™πŸ + 𝟏𝟐𝟐 π’™πŸ = πŸπŸ”πŸ— βˆ’ πŸπŸ’πŸ’

π’™πŸ = πŸπŸ“ 𝒙 = πŸ“

RIGHT TRIANGLE TRIGONOMETRY