Glossary

3.3 Trigonometric Functions of Any Angle

2 min read•july 25, 2024

extend beyond the basic right triangle, applying to all angles in the . This broader view allows us to work with angles of any size, even those greater than 360° or negative angles.

Understanding how trig functions behave in different quadrants is crucial. The helps remember which functions are positive where, while reference angles simplify calculations for non-standard angles. These concepts are key to mastering trigonometry.

Understanding Trigonometric Functions in All Quadrants

Trigonometric functions in all quadrants

Top images from around the web for Trigonometric functions in all quadrants

  • Unit Circle: A circle with a radius of 1. View original

    Is this image relevant?

  • Trigonometric Functions and the Unit Circle | Boundless Algebra View original

    Is this image relevant?

  • Unit Circle: A circle with a radius of 1. View original

    Is this image relevant?

1 of 3

Top images from around the web for Trigonometric functions in all quadrants

  • Unit Circle: A circle with a radius of 1. View original

    Is this image relevant?

  • Trigonometric Functions and the Unit Circle | Boundless Algebra View original

    Is this image relevant?

  • Unit Circle: A circle with a radius of 1. View original

    Is this image relevant?

1 of 3
  • Unit circle defines trig functions for all real numbers allows angles > 360° or < 0° (full rotations)
  • Coterminal angles share terminal side differ by multiples of 360° (720°720° and 0°0°)
  • repeats every 360° (2π radians) (sin⁡θ=sin⁡(θ+360°)\sin \theta = \sin (\theta + 360°))
  • unique properties for trig functions (Q1: all positive, Q2: only positive)

Signs of trigonometric functions

  • ASTC rule All positive in Q1, Sine in Q2, in Q3, in Q4
  • follow primary functions ( follows sine)
  • x and y coordinates sine relates to y-coordinate cosine to x-coordinate
  • Tangent positive when sine and cosine have same sign negative when opposite (tan⁡45°>0\tan 45° > 0, tan⁡225°<0\tan 225° < 0)

Reference angles for evaluation

  • acute angle with x-axis always positive ≤ 90°
  • :
    1. Q1: θ
    2. Q2: 180° - θ
    3. Q3: θ - 180°
    4. Q4: 360° - θ
  • Using reference angles absolute value equals reference angle apply quadrant sign (sin⁡150°=sin⁡30°\sin 150° = \sin 30°)

Unit circle for problem-solving

  • Unit circle radius 1 unit center at origin (0, 0)
  • Coordinates x = cosθ, y = sinθ
  • 30°, 45°, 60° and multiples memorize for efficiency (cos⁡30°=32\cos 30° = \frac{\sqrt{3}}{2})
  • θrad=θdeg×π180°θ_{rad} = θ_{deg} × \frac{π}{180°}, θdeg=θrad×180°πθ_{deg} = θ_{rad} × \frac{180°}{π}
  • Solving triangles find missing sides or angles using unit circle values
  • plot key points using unit circle (sin⁡90°=1\sin 90° = 1, cos⁡180°=−1\cos 180° = -1)

Key Terms to Review (19)

30 degrees: 30 degrees is a specific angle measurement commonly encountered in trigonometry, representing one-sixth of a full rotation. This angle is fundamental in various trigonometric calculations, allowing for the evaluation of sine, cosine, and tangent values, as well as serving as a building block for understanding angles in different contexts such as the unit circle and right triangles.
45 degrees: 45 degrees is an angle that is exactly halfway between 0 degrees and 90 degrees, making it one of the most commonly encountered angles in trigonometry. This angle is significant because it has specific sine, cosine, and tangent values, and it plays a crucial role in understanding the behavior of trigonometric functions in various contexts.
60 degrees: 60 degrees is a specific angle measure that is significant in trigonometry, especially in the study of triangles and circular functions. It is one of the angles in an equilateral triangle, where all angles are equal, and each measures 60 degrees. Understanding 60 degrees is crucial for calculating sine, cosine, and tangent values, as well as for applying sum and difference identities in trigonometric equations.
Angle Measure Conversions: Angle measure conversions involve changing the measurement of an angle from one unit to another, typically from degrees to radians or vice versa. This concept is crucial in trigonometry, particularly when dealing with the trigonometric functions of any angle, as different contexts may require angles to be in specific units for calculations or interpretations.
ASTC Rule: The ASTC Rule is a mnemonic device used to remember the signs of trigonometric functions in each of the four quadrants of the Cartesian coordinate system. It stands for 'All Students Take Calculus,' indicating that in the first quadrant, all trigonometric functions are positive, in the second quadrant only sine and its reciprocal function cosecant are positive, in the third quadrant only tangent and its reciprocal function cotangent are positive, and in the fourth quadrant only cosine and its reciprocal function secant are positive.
Calculating Reference Angles: Calculating reference angles involves finding the smallest angle formed by the terminal side of a given angle and the x-axis. This concept is crucial for understanding how angles are represented in different quadrants and is essential for evaluating trigonometric functions for angles beyond the standard range of 0 to 360 degrees or 0 to 2π radians.
Coordinates of points: Coordinates of points are ordered pairs or triplets that define the position of a point in a given space, typically represented as (x, y) in two dimensions or (x, y, z) in three dimensions. Understanding these coordinates is essential for graphing trigonometric functions, evaluating angles, and applying identities effectively. They help visualize relationships between angles and their corresponding values, which is crucial for solving various mathematical problems.
Cosecant: Cosecant is a trigonometric function that is defined as the reciprocal of the sine function. This means that for any angle, the cosecant can be calculated by taking 1 divided by the sine of that angle. It plays a crucial role in understanding various aspects of trigonometry, including relationships in right triangles, circular functions, and fundamental identities.
Cosine: Cosine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. This relationship holds true in various contexts, from defining circular functions on the unit circle to solving complex triangles, making it essential in understanding angles and their measures.
Graphing Trig Functions: Graphing trig functions involves plotting the sine, cosine, tangent, and their reciprocal functions on a coordinate plane to visualize their periodic nature and behavior over various intervals. Understanding how these functions behave helps in solving real-world problems, analyzing cycles, and interpreting data in fields such as physics and engineering. Each function has distinct characteristics like amplitude, period, and phase shift, which all affect its graphical representation.
Periodic Nature: The periodic nature refers to the repeating pattern of values that trigonometric functions exhibit as angles increase or decrease. This characteristic allows these functions, like sine and cosine, to have outputs that cycle through a predictable range as the angle moves through its full rotation, enabling the modeling of phenomena that exhibit similar repetitive behaviors.
Quadrant-specific characteristics: Quadrant-specific characteristics refer to the unique properties and behaviors of trigonometric functions based on the quadrant in which an angle resides. These characteristics help in determining the signs of sine, cosine, and tangent values depending on whether the angle is located in the first, second, third, or fourth quadrant, and are essential for evaluating trigonometric functions of any angle accurately.
Reciprocal Functions: Reciprocal functions are mathematical functions that are defined as the multiplicative inverse of another function. In trigonometry, the most common reciprocal functions are sine, cosine, and tangent's inverses: cosecant (csc), secant (sec), and cotangent (cot). These functions are essential in understanding the relationships between angles and sides of triangles, particularly in relation to the unit circle and their graphical representations.
Reference Angle: A reference angle is the smallest angle formed by the terminal side of a given angle and the x-axis, always measured as a positive acute angle. This concept is crucial for understanding how angles relate to trigonometric functions across different quadrants, as it helps simplify calculations by connecting them to familiar angles in the first quadrant.
Sine: Sine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. This relationship extends beyond right triangles to various contexts, such as the unit circle and periodic functions, where sine represents the y-coordinate of a point on the unit circle corresponding to a given angle.
Special Angles: Special angles refer to specific angles that have well-known sine, cosine, and tangent values, making them easier to work with in trigonometric calculations. These angles include 0°, 30°, 45°, 60°, and 90°, and they form the foundation for understanding the behavior of trigonometric functions. Their simplicity and regularity allow for quick mental calculations and serve as reference points in the unit circle.
Tangent: The tangent function, often abbreviated as 'tan', relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side. This function is crucial in various branches of mathematics, especially in solving triangles and modeling periodic phenomena.
Trigonometric Functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are foundational in studying relationships in right triangles and can be extended to any angle, making them crucial for various applications in mathematics, physics, and engineering. Understanding these functions allows for the analysis of periodic phenomena and is essential for solving real-world problems involving angles and distances.
Unit circle: The unit circle is a circle with a radius of one centered at the origin of a coordinate system, commonly used to define trigonometric functions. It serves as a fundamental tool for visualizing the relationships between angles and their corresponding sine and cosine values, helping to extend trigonometric functions to all real numbers and providing insights into complex numbers and their roots.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide