Understanding Trigonometry

Understanding Trigonometry

• Trigonometry deals with triangles, particularly right-angled triangles.

Sine, Cosine, and Tangent

• Sine (sin) of an angle is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) of an angle is the ratio of the opposite side to the adjacent side.

Trigonometric Functions as Ratios

• Trigonometric functions are ratios of sides of a right-angled triangle.
• These ratios do not depend on the size of the triangle, but only on the angle.

Trigonometric Functions as Functions

• Trigonometric functions take an angle as input and return a value.
• This perspective allows for the definition of trigonometric functions for angles beyond 0 to 90 degrees.

Unit Circle

• The unit circle is a circle with a radius of 1 unit.
• Trigonometric functions can be defined in terms of the unit circle, where the input angle corresponds to a point on the circle.
• The x-coordinate of the point corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle.

Calculators and Trigonometric Functions

• Calculators can be used to calculate trigonometric functions for a given angle.
• Calculators may return multiple values for a single input angle, due to the periodic nature of trigonometric functions.

Importance of Understanding Trigonometry

• Understanding trigonometry is important for solving problems that involve right-angled triangles.
• Trigonometry has many real-world applications, including physics, engineering, and computer science.

Understanding Trigonometric Functions

• Trigonometric functions are initially defined in terms of right triangles, but later extended to work with angles greater than 90 degrees.
• The concept of trigonometric functions is crucial to understand, as it involves calculating angles and using the inverse functions.

Using the Calculator

• When using the calculator to find the angle, it returns the value of alpha (α) in the first quadrant, between 0 and π/2 radians.
• The calculator returns the value of beta (β) between 0 and π radians, which is in the first or second quadrant.
• The value of gamma (γ) can be anywhere between -π and π radians.

Importance of Understanding

• Understanding the concept of trigonometric functions is essential to avoid mistakes and to be able to calculate angles correctly.
• It's not just about applying formulas, but about using logic and understanding the concepts to find the correct answers.

Applying Inverse Functions

• The inverse functions (arcsin, arccos, arctan) are used to find the angle, but they have limitations and may not always return the correct answer.
• The calculator returns the value of alpha (α) using the inverse function arcsin, but it may not return the correct answer for all angles.
• Using the inverse function arccos returns the value of beta (β) between 0 and π radians, and arctan returns the value of gamma (γ) between -π and π radians.

Conclusion

• Understanding trigonometric functions is crucial to calculate angles correctly.
• It's essential to understand the concepts behind the formulas and not just apply them blindly.