Question

Give the algebraic signs of the sine, cosine, and tangent of the following. Do not use your calculator. $$110^{\circ}$$

Short answer

For \(110^\circ\), sine is positive, cosine is negative, and tangent is negative.

Step-by-step solution

Determine the quadrant

To figure out the signs of the trigonometric functions, first identify the quadrant where the angle lies. Angles from \(90^\circ\) to \(180^\circ\) lie in the second quadrant.

Identify the signs of trigonometric functions in the second quadrant

In the second quadrant, sine is positive, cosine is negative, and tangent is negative since tangent is the ratio of sine to cosine.

Key concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right triangles. But it doesn't stop there! Trigonometry also delves into the properties of waves, such as sound and light, which resemble the trigonometric functions.

When we discuss angles in trigonometry, we often refer to the functions like sine, cosine, and tangent. Each function represents a ratio of sides in a right triangle when the angle is known (sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent). These functions also give us the y-coordinate, x-coordinate, and slope of the terminal side of an angle within the unit circle, respectively.

The significance of knowing the algebraic signs of trigonometric functions at various angles cannot be overstated. It allows us to predict the behavior of these functions without requiring precise numerical values, making it a powerful tool for solving geometric problems.

Significance in Real-world Applications

For instance, engineers and architects might use the properties of trigonometric functions to calculate forces in structures, while physicists employ them in wave theory. Trigonometry is everywhere— from the ebb and flow of tides to the design of intricate video game graphics.

Quadrantal Angles and Their Characteristics

Angles that are multiples of 90 degrees, such as 0, 90, 180, 270, and 360 degrees, are known as quadrantal angles. These angles are special because they fall directly on the x or y-axis when portrayed in the coordinate system on the unit circle.

At these quadrantal positions, sine and cosine typically take on values of 0, 1, or -1. However, tangent may not be defined (when the angle results in division by zero). Understanding the behavior of trigonometric functions at quadrantal angles is integral to grasping their overall properties and can simplify complex trigonometric problems into manageable slices.

Exercise Improvement Advice

When improving your understanding of the trigonometric function signs at various angles, like those in this exercise, it is essential to visualize them within the coordinate plane and remember the ASTC rule— which stands for All, Sine, Tangent, Cosine— to recall which functions are positive in which quadrants. This mnemonic device corresponds to the four quadrants of the Cartesian plane in counterclockwise order, starting from the positive x-axis.

The Unit Circle: A Fundamental Tool in Trigonometry

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate system. It is an incredibly useful tool in trigonometry for visualizing angular positions and the corresponding values of trigonometric functions.

The x-coordinate of any point on the unit circle corresponds to the cosine of the angle that point makes with the positive x-axis, and the y-coordinate corresponds to the sine of that angle. For angles within the unit circle, their cosine and sine values are simply the coordinates of the point where the terminal side of the angle intersects the circle.

Applications of the Unit Circle

As angles increase past 90 degrees and continue counterclockwise around the unit circle, you can infer the signs of the trigonometric functions based on the quadrant in which they land. For instance, in our exercise considering the angle of 110 degrees, it's located in the second quadrant, where the x-coordinate (cosine) is negative, and the y-coordinate (sine) is positive, consistent with what was found in the step-by-step solution.