TMAT 103

TMAT 103

Chapter 3

Right-Triangle Trigonometry

TMAT 103

§3.1

The Trigonometric Ratios

§3.1 – The Trigonometric Ratios

• Angle– Initial side– Terminal side– Vertex

§3.1 – The Trigonometric Ratios

• Measurement– Degrees– Minutes– Seconds

§3.1 – The Trigonometric Ratios

•Measurement Tools - Protractor

§3.1 – The Trigonometric Ratios

• Types of angles– Obtuse

• Greater than 90°

– Acute• Less than 90°

– Right• Exactly 90°

§3.1 – The Trigonometric Ratios

• Converting minutes and seconds to degrees– 1’ = 1/60°– 1’’ = 1/3600°– To convert x° y’ z’’ to degrees:

°

x° y’ z’’ =

– Ex: Convert 33° 14’ 52’’ to degrees

360060

zyx

§3.1 – The Trigonometric Ratios

• Converting degrees to minutes, and seconds– Reverse the procedure on the previous slide– To convert x.y to minutes and seconds:

• x is the degrees

• Multiply y by 60– The integer part is the minutes

– Multiply the fractional part by 60 – the new integer part is the seconds

– Ex: Convert 33.71 to degrees, minutes, and seconds

§3.1 – The Trigonometric Ratios

• Pythagorean Theorem (Right triangles)c2 = a2 + b2

§3.1 – The Trigonometric Ratios

• Ex: Find c in the diagram below

§3.1 – The Trigonometric Ratios

• Ex: Find a in the diagram below

§3.1 – The Trigonometric Ratios• Trigonometric ratios

– Relationship between an acute angle of a right triangle and the lengths of its sides

• sin A = side opposite A hypotenuse

• cos A = side adjacent to A hypotenuse

• tan A = side opposite A side adjacent to A

• cot A = side adjacent to A side opposite A

• sec A = hypotenuse side adjacent to A

• csc A = hypotenuse side opposite A

§3.1 – The Trigonometric Ratios

• Ex: Find the 6 trigonometric ratios for A

§3.1 – The Trigonometric Ratios

• Similar triangles– Corresponding angles are equal– Consider sin A in the picture below

• Does it matter if ABC is used, or ADE?

§3.1 – The Trigonometric Ratios

• Examples– Show that cot A = cos A by using the ratios

sin A

– Given that sin A = .8387, and cos A = .5446, compute the remaining 4 trigonometric ratios for A. Round to 4 significant figures.

TMAT 103

§3.2

Values of the Trigonometric Ratios

§3.2 – Values of the Trigonometric Ratios

• Trigonometric ratios of the 60° and 30° angles.

§3.2 – Values of the Trigonometric Ratios

• Trigonometric ratios of the 45° angle.

§3.2 – Values of the Trigonometric Ratios

• Trigonometric ratios of the other angles.– Use a calculator

• Examples:– Find sin 65.25°– Find cot 22° 3’ 44’’– Find if cos = 0.5402

TMAT 103

§3.3

Solving Right Triangles

§3.3 – Solving Right Triangles

• Solving a triangle – Finding unknown values of sides or angles

• Tools needed to solve triangles– Pythagorean theorem– Complementary angles add to 90°– Trigonometric ratios

§3.3 – Solving Right Triangles

• Ex: Find angle A to the nearest hundredth of a degree.

§3.3 – Solving Right Triangles

• Ex: Completely solve the given triangle.

TMAT 103

§3.4

Applications of the Right Triangle

§3.4 – Applications of the Right Triangle

• A pilot is piloting a helicopter in the wilderness at 1200 feet about the ground searching for a downed plane. As the pilot spots it, she measures its angle of depression as 53°. She also spots a road whose angle of depression is 15°. Find the distance of the downed plane to the road.

§3.4 – Applications of the Right Triangle

• The pathway for a mineshaft is described as follows: The first shaft extends north and down at an angle of 3.5° for 225 feet to a vertical shaft which is 125 feet long. A third shaft then extends north and down at an angle of 6.8° for 175 feet. What is the total depth below ground level at the end of the third shaft? Also, what is the net horizontal distance from the ground opening to the end of the third shaft?