Evaluating the Expression 4sin2t tan t at t 495°: A Comprehensive Guide for SEO Optimal Content
Understanding trigonometric functions and their values can be crucial in a variety of mathematical and engineering applications. This guide delves into the process of evaluating the expression 4sin2t tan t at t 495°, breaking down the steps and providing key concepts to improve SEO for this topic.
Introduction to Trigonometry and Trigonometric Functions
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. The fundamental trigonometric functions, sine (sin) and tangent (tan), are of primary importance. These functions are defined based on the angles in a right-angled triangle, and their values can be determined using reference angles and the unit circle.
Navigating the Unit Circle
The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It is a powerful tool for understanding trigonometric functions. Each point on the circle corresponds to an angle in standard position and its associated sine, cosine, and tangent values.
Evaluating 4sin2t tan t at t 495°
To evaluate the expression 4sin2t tan t at t 495°, we need to simplify the angle and then apply the trigonometric identities.
Step 1: Simplify the Angle
Angles in trigonometry can be reduced to an equivalent angle within the standard range of 0° to 360°. This is achieved by subtracting or adding 360° as many times as necessary until the angle lies within this range.
Given t 495°, we can simplify it as follows:
495° - 360° 135°
Step 2: Calculate sin2135°
The value of sin 135° can be determined using the reference angle 45°. In the second quadrant, the sine function is positive.
sin 135° sin (180° - 45°) sin 45° √2/2
Therefore, we calculate:
sin2135° (sin 135°)2 (√2/2)2 2/4 1/2
Step 3: Calculate tan 135°
In the second quadrant, the tangent function is negative. The reference angle for 135° is 45°.
tan 135° -tan 45° -1
Step 4: Combine the Values
Substituting the values back into the expression:
4sin2135° tan 135° 4 × (1/2) × (-1) 4 × -1/2 -2
Conclusion
Thus, the value of the expression 4sin2495° tan 495° is -2. This process demonstrates the importance of simplifying angles and using trigonometric identities to evaluate expressions.
Frequently Asked Questions (FAQ)
What is the value of 4sin2t tan t when t 495°?
By simplifying the angle and using trigonometric identities, the value of 4sin2t tan t at t 495° is -2.
How do you use the unit circle to find trigonometric values?
The unit circle allows you to find the sine, cosine, and tangent values for any angle by using reference angles and the signs in each quadrant. In the second quadrant, the sine function is positive, and the tangent function is negative.
What are the key trigonometric functions?
The key trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions are essential in evaluating expressions and solving problems in trigonometry.