Which Trig Functions Are Positive per Quadrant Practice
Use quadrant signs to identify positive trigonometric functions
Which Trig Functions Are Positive per Quadrant Practice
Use quadrant signs to identify positive trigonometric functions
Math - Grade 9-12
- 1
In Quadrant I, which of the six trigonometric functions are positive?
Quadrant I is the quadrant where both coordinates on the unit circle are positive.
In Quadrant I, all six trigonometric functions are positive: sine, cosine, tangent, cosecant, secant, and cotangent. This is because both x and y are positive. - 2
In Quadrant II, which trigonometric functions are positive?
In Quadrant II, sine and cosecant are positive. The y-coordinate is positive and the x-coordinate is negative, so sine is positive, cosine is negative, and tangent is negative. - 3
In Quadrant III, which trigonometric functions are positive?
A negative divided by a negative is positive.
In Quadrant III, tangent and cotangent are positive. Both x and y are negative, so sine and cosine are negative, but their ratio is positive. - 4
In Quadrant IV, which trigonometric functions are positive?
In Quadrant IV, cosine and secant are positive. The x-coordinate is positive and the y-coordinate is negative, so cosine is positive and sine is negative. - 5
An angle θ is in Quadrant II. Determine the signs of sin θ, cos θ, and tan θ.
Use sin θ = y, cos θ = x, and tan θ = y/x on the unit circle.
For an angle in Quadrant II, sin θ is positive, cos θ is negative, and tan θ is negative. This is because y is positive, x is negative, and y divided by x is negative. - 6
An angle θ is in Quadrant III. Determine the signs of csc θ, sec θ, and cot θ.
For an angle in Quadrant III, csc θ is negative, sec θ is negative, and cot θ is positive. Cosecant has the same sign as sine, secant has the same sign as cosine, and cotangent has the same sign as tangent. - 7
Use the ASTC rule to complete the statement: In Quadrant IV, only the functions related to which trig function are positive?
ASTC can stand for All, Sine, Tangent, Cosine as you move counterclockwise from Quadrant I.
In Quadrant IV, only the functions related to cosine are positive. Cosine and its reciprocal, secant, are positive in Quadrant IV. - 8
An angle has sin θ > 0 and cos θ < 0. In which quadrant is θ located, and which reciprocal trig function is also positive?
The angle θ is in Quadrant II because sine is positive and cosine is negative there. The reciprocal trig function that is also positive is cosecant. - 9
An angle has tan θ > 0 and sin θ < 0. In which quadrant is θ located? Explain your reasoning.
Find the quadrant that satisfies both sign statements at the same time.
The angle θ is in Quadrant III. Tangent is positive in Quadrants I and III, but sine is negative only in Quadrants III and IV, so both conditions are true in Quadrant III. - 10
For an angle θ in Quadrant IV, decide whether each statement is true or false: sin θ < 0, sec θ > 0, cot θ > 0.
The statement sin θ < 0 is true, sec θ > 0 is true, and cot θ > 0 is false. In Quadrant IV, sine is negative, secant is positive, and cotangent is negative. - 11
A point on the terminal side of an angle θ has coordinates (-5, -12). Which quadrant is the angle in, and which trigonometric functions are positive?
Use the signs of x and y to identify the quadrant first.
The angle is in Quadrant III because both coordinates are negative. In Quadrant III, tangent and cotangent are positive. - 12
A student says, "Cosine is positive in Quadrant II because Quadrant II is above the x-axis." Explain the mistake and give the correct sign of cosine in Quadrant II.
Cosine matches the sign of x, while sine matches the sign of y.
The mistake is using the y-coordinate to decide the sign of cosine. Cosine depends on the x-coordinate, and x is negative in Quadrant II, so cosine is negative in Quadrant II.