Determine the signs of the trigonometric functions of an angle in standard position with the given measure. SeeExample 2. 178°

Verified step by step guidance

1. Identify the quadrant in which the angle 178° lies. Since 178° is between 90° and 180°, it is in the second quadrant.
2. Recall the ASTC rule (All Students Take Calculus) which helps determine the sign of trigonometric functions in each quadrant: All functions are positive in the first quadrant, Sine is positive in the second, Tangent is positive in the third, and Cosine is positive in the fourth.
3. Since 178° is in the second quadrant, the sine function is positive, while the cosine and tangent functions are negative.
4. For the cosecant, secant, and cotangent functions, remember they are the reciprocals of sine, cosine, and tangent respectively. Therefore, cosecant is positive, while secant and cotangent are negative.
5. Summarize: In the second quadrant, for an angle of 178°, sine and cosecant are positive, while cosine, secant, tangent, and cotangent are negative.

Key Concepts

Standard Position of an Angle

An angle is in standard position when its vertex is at the origin of a coordinate system and its initial side lies along the positive x-axis. The angle is measured counterclockwise from the initial side. For example, an angle of 178° is located in the second quadrant, where the terminal side is positioned slightly past the negative x-axis.

Quadrants and Signs of Trigonometric Functions

The coordinate plane is divided into four quadrants, each with distinct signs for the trigonometric functions. In the first quadrant, all functions are positive; in the second, sine is positive while cosine and tangent are negative; in the third, tangent is positive while sine and cosine are negative; and in the fourth, cosine is positive while sine and tangent are negative. Understanding these signs is crucial for determining the values of trigonometric functions based on the angle's quadrant.

Reference Angle

The reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is always positive and helps in determining the values of trigonometric functions. For an angle of 178°, the reference angle is 180° - 178° = 2°, which can be used to find the sine, cosine, and tangent values, considering the signs based on the quadrant.