Trigonometric Functions: A Comprehensive Guide with Exercises, Exercises of Mathematics

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CHAPTER 5

TRIGONOMETRIC

FUNCTIONS

TRIGONOMETRIC FUNCTIONS

• Angle measure
• Trigonometric of Right Angle
• Trigonometric Functions of Angles
• Trigonometric Functions of Real Numbers
• Trigonometric Graphs

Definition : Degree Measure of Angles

An angle formed by one complete counterclockwise rotation has measure 360 degrees

360 ° represents 1 complete rotation

180 ° represents

1 2

rotation

90 ° represents

1 4

rotation

Types of Angle

Right Angle An angle measuring exactly 90 ° Straight Angle An angle measuring exactly 180 ° Acute Angle An angle measuring greater than 0 ° but less than 90 ° Obtuse Angle An angle measuring greater than 90 ° but less than 180 °

Relationship between degrees and radians

1. To convert degrees to radians, multiply by 𝜋 180

180 ° = 𝜋 rad 1 rad =

°

rad

2. To convert radians to degrees, multiply by 180 𝜋

Definition of Radian Measure

One radian is a measure of the central angle of a circle subtended by an arc equal in length to the radius of the circle

Example :

1. Express 60 ° in radians

Solution

2. Express

𝜋 6

rad in degrees

Solution

TRIGONOMETRY OF RIGHT TRIANGLES

sin 𝜃 =

cos 𝜃 =

tan 𝜃 =

csc 𝜃 =

sec 𝜃 =

cot 𝜃 =

𝜃 adjacent, 𝒂 opposite, 𝒃

THE PHYTAGOREAN IDENTITIES

𝜃 adjacent, 𝒂 opposite, 𝒃

2

2

2

2

2

2

sin 𝜃

2

+ cos 𝜃

2

2

sin

2

𝜃 + cos

2

𝜃 = 1 ⟶^ (^1 )

tan

2

𝜃 + 1 = s𝑒𝑐

2

1 + cot

2

2

Example 2 :

A 40 𝑓𝑡 ladder leans against a building. If the base of the ladder is 6 𝑓𝑡 from the base of the building, what is the angle formed by the ladder and the building. Solution

Example 3 :

From a point on the ground 500 𝑓𝑡 from the base of a building, it is observed that the angle of elevation to the top of the building is 24 ° and the angle of elevation to the top of a flagpole atop the building is 27 °. Find the height of the building and the length of the flagpole.

EVEN-ODD PROPERTIES

Even functions are functions for which 𝑓 −𝑥 = 𝑓(𝑥) and symmetric about 𝑦 −axis Odd functions are functions for which 𝑓 −𝑥 = −𝑓(𝑥) and symmetric about the origin Even functions

cos −𝑥 = cos 𝑥 sec^ −𝑥^ =^ sec^ 𝑥

Odd functions

sin(−𝑥) = − sin 𝑥 csc(−𝑥) = − csc 𝑥

tan(−𝑥) = − tan 𝑥 cot(−𝑥) = − cot 𝑥

Example 1 : Find each value

1 ) sin(−

Solution

2 ) cos −

Solution

Example 3 :

Write tan 𝑡 in terms of cos 𝑡, where 𝑡 is in Quadrant III.

Example 4 :

If sec 𝜃 = 2 and 𝜃 is in Quadrant IV. Find the other trigonometric functions of 𝜃.