Podcast
Questions and Answers
What is the value of $ an rac{ heta}{4}$ if $ heta = 0°$?
What is the value of $ an rac{ heta}{4}$ if $ heta = 0°$?
- 3
- 0 (correct)
- Undefined
- 1
Which of the following statements regarding sine and cosine functions is correct?
Which of the following statements regarding sine and cosine functions is correct?
- cos(-x) = -cos(x)
- sin(-x) = sin(x)
- cos(-x) = cos(x) (correct)
- sin(x) is always positive
What is the range of the cosine function?
What is the range of the cosine function?
- 0 to 1
- 0 to $ rac{ heta}{2}$
- -1 to 0
- -1 to 1 (correct)
What is the reciprocal of sin x?
What is the reciprocal of sin x?
In which quadrant are both sine and cosine values negative?
In which quadrant are both sine and cosine values negative?
What is the radius of the circle when a central angle of 60° intercepts an arc of length 37.4 cm?
What is the radius of the circle when a central angle of 60° intercepts an arc of length 37.4 cm?
How far does the tip of the minute hand of a watch move in 40 minutes if the length of the minute hand is 1.5 cm?
How far does the tip of the minute hand of a watch move in 40 minutes if the length of the minute hand is 1.5 cm?
What is the relationship between the radii of two circles if the arcs of the same lengths subtend angles of 65° and 110° respectively?
What is the relationship between the radii of two circles if the arcs of the same lengths subtend angles of 65° and 110° respectively?
Convert 25° to radians. What is the correct measure?
Convert 25° to radians. What is the correct measure?
What is the radian measure of –47°30′?
What is the radian measure of –47°30′?
What is the radian conversion of 240°?
What is the radian conversion of 240°?
In how many minutes does the minute hand make a complete revolution?
In how many minutes does the minute hand make a complete revolution?
What would the length of an arc be in a circle with a radius of 3 cm and an angle of 90°?
What would the length of an arc be in a circle with a radius of 3 cm and an angle of 90°?
What is the value of cos(π)?
What is the value of cos(π)?
Which of the following statements is true regarding the sine function?
Which of the following statements is true regarding the sine function?
For which values of x does cos(x) equal zero?
For which values of x does cos(x) equal zero?
What relationship holds true for all real x concerning sine and cosine?
What relationship holds true for all real x concerning sine and cosine?
When is sec x defined?
When is sec x defined?
What is the value of sin(3π)?
What is the value of sin(3π)?
What is the value of tan(π/4)?
What is the value of tan(π/4)?
For what integer multiples does tan(x) become undefined?
For what integer multiples does tan(x) become undefined?
What does the term 'trigonometry' literally mean?
What does the term 'trigonometry' literally mean?
Which profession is NOT mentioned as having historically used trigonometry?
Which profession is NOT mentioned as having historically used trigonometry?
In which of the following applications is trigonometry used?
In which of the following applications is trigonometry used?
What does a positive angle represent?
What does a positive angle represent?
What is the initial side of an angle?
What is the initial side of an angle?
What is the vertex in the context of an angle?
What is the vertex in the context of an angle?
Which of the following units are commonly used for measuring angles?
Which of the following units are commonly used for measuring angles?
What is indicated by the amount of rotation in angle measurement?
What is indicated by the amount of rotation in angle measurement?
What is the radian measure of an angle measuring 60°?
What is the radian measure of an angle measuring 60°?
Which formula correctly relates degree and radian measures?
Which formula correctly relates degree and radian measures?
How many degrees are in $ rac{3 heta}{ heta}$ radians?
How many degrees are in $ rac{3 heta}{ heta}$ radians?
What is the degree equivalent of $ rac{7 heta}{6}$ radians?
What is the degree equivalent of $ rac{7 heta}{6}$ radians?
What is the radian measure of an angle that is 270°?
What is the radian measure of an angle that is 270°?
If 40° 20′ is converted into radians, what is its equivalent?
If 40° 20′ is converted into radians, what is its equivalent?
Which of the following radian measures corresponds to 180°?
Which of the following radian measures corresponds to 180°?
What is the degree measure corresponding to 6 radians?
What is the degree measure corresponding to 6 radians?
Study Notes
Trigonometric Functions
- Trigonometry is the study of the relationships between the sides and angles of triangles.
- Angles are measured in degrees or radians.
- Degree measure divides a circle into 360 degrees (360°).
- Radian measure defines an angle as the ratio of the arc length to the radius of a circle.
- Conversion formulas:
- Radian measure = (180/π) × Degree measure.
- Degree measure = (π/180) × Radian measure.
- Quadrantal angles are angles whose terminal side coincides with one of the axes. Examples include 0°, 90°, 180°, and 270°.
- Trigonometric functions are functions that relate an angle to the ratios of the sides of a right-angled triangle.
- Basic trigonometric functions:
- Sine (sin): Opposite side / Hypotenuse
- Cosine (cos): Adjacent side / Hypotenuse
- Tangent (tan): Opposite side / Adjacent side
- Cosecant (csc): 1 / Sin
- Secant (sec): 1 / Cos
- Cotangent (cot): 1 / Tan
Trigonometric Identities
- Fundamental trigonometric identity: sin²x + cos²x = 1
- Other identities:
- 1 + tan²x = sec²x
- 1 + cot²x = csc²x
Values of Trigonometric Functions
- The values of trigonometric functions for certain common angles, such as 0°, 30°, 45°, 60°, and 90°, can be determined using the unit circle.
Sign of Trigonometric Functions
- The sign of a trigonometric function depends on the quadrant in which the angle lies.
- First Quadrant (0° < x < 90°): All trigonometric functions are positive.
- Second Quadrant (90° < x < 180°): Sine and cosecant are positive; others are negative.
- Third Quadrant (180° < x < 270°): Tangent and cotangent are positive; others are negative.
- Fourth Quadrant (270° < x < 360°): Cosine and secant are positive; others are negative.
- Key formulas:
- cos (-x) = cos x
- sin (-x) = -sin x
- -1 ≤ cos x ≤ 1 and -1 ≤ sin x ≤ 1 for all x.
Periodicity of Trigonometric Functions
- The values of sine and cosine functions repeat after a full cycle of 2π (360°).
- sin(2nπ + x) = sin x, n ∈ Z
- cos(2nπ + x) = cos x, n ∈ Z.
Special Cases
- sin x = 0 implies x = nπ, where n is any integer.
- cos x = 0 implies x = (2n + 1)(π/2), where n is any integer.
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Description
Test your knowledge of trigonometric functions and their relationships to angles and triangles. This quiz covers concepts like degree and radian measures, as well as the basic trigonometric functions such as sine, cosine, and tangent.
