What is sin(π/2)? — 1.At π/2 (90°), the unit circle point is (0, 1). Since sin = y-coordinate, sin(π/2) = 1.
What is the radius of the unit circle? — 1.By definition, the unit circle has radius exactly 1, centered at the origin. "Unit" refers to the length of the radius being one unit.
In the unit circle, what is the x-coordinate at angle θ equal to? — cos θ.For a unit circle, the terminal point at angle θ has coordinates (cos θ, sin θ). The x-coordinate = cos θ.
What is tan(π/4)? — 1.tan(π/4) = sin(π/4)/cos(π/4) = (√2/2)/(√2/2) = 1. Or simply: at 45°, the angle is equal in both components.
In which quadrant is the angle 5π/4 (225°)? — Q3.5π/4 = 225°, which is between 180° and 270° — that is Quadrant III (bottom-left).
What is cos(2π)? — 1.2π = 360°, which brings you back to the starting point (1, 0) on the unit circle. So cos(2π) = 1.
What is the period of the sine function? — 2π.The sine (and cosine) function completes one full cycle in 2π radians (360°). Its period is 2π.
What is cos(π)? — −1.At π (180°), the unit circle point is (−1, 0). So cos(π) = −1.
What is the radian measure of one full revolution? — 2π.One full revolution = 360° = 2π radians. This is because the circumference of the unit circle is 2π·1 = 2π.
What is tan(0)? — 0.tan(0) = sin(0)/cos(0) = 0/1 = 0. The tangent is 0 wherever the sine is 0 (and cosine ≠ 0).
The amplitude of y = A sin(x) is: — |A|.Amplitude = |A|. The absolute value ensures amplitude is always positive regardless of reflection.
The period of y = sin(Bx) is: — 2π/B.Period = 2π/|B|. Larger B compresses the wave; smaller B stretches it.
What is the period of y = cos(3x)? — 2π/3.Period = 2π/|B| = 2π/3.
What is the amplitude of y = −4 sin(x)? — 4.Amplitude = |−4| = 4. The negative sign reflects the graph but doesn't change amplitude.
The vertical shift D in y = A sin(Bx) + D represents: — The midline y = D.D shifts the graph vertically. The midline of the wave is y = D.
For y = sin(x − π/2), the phase shift is: — Right π/2.y = sin(x − π/2): the shift is C = π/2 to the RIGHT. (x − C) → right shift by C.
The Pythagorean identity states: — sin²θ + cos²θ = 1.The fundamental Pythagorean identity is sin²θ + cos²θ = 1, derived from x² + y² = 1 on the unit circle.
csc θ equals: — 1/sin θ.csc θ = 1/sin θ (cosecant is the reciprocal of sine).
sec θ equals: — 1/cos θ.sec θ = 1/cos θ (secant is the reciprocal of cosine).
tan θ in terms of sin and cos: — sin θ/cos θ.tan θ = sin θ/cos θ. This is the quotient identity for tangent.
Which identity follows from dividing sin²θ + cos²θ = 1 by cos²θ? — tan²θ + 1 = sec²θ.Dividing each term by cos²θ: tan²θ + 1 = sec²θ.
sin(2θ) equals: — 2sinθ cosθ.The double-angle formula: sin(2θ) = 2 sin θ cos θ.
cot θ equals: — cos θ/sin θ.cot θ = cos θ/sin θ = 1/tan θ. Cotangent is the reciprocal of tangent.
Simplify: sin θ · csc θ — 1.csc θ = 1/sin θ, so sin θ · csc θ = sin θ · (1/sin θ) = 1.
Simplify: (tan θ)(cot θ) — 1.tan θ = sin θ/cos θ and cot θ = cos θ/sin θ, so their product = 1.
sin(A + B) equals: — sinA cosB + cosA sinB.sin(A + B) = sinA cosB + cosA sinB. The signs match the operation.
cos(A + B) equals: — cosA cosB − sinA sinB.cos(A + B) = cosA cosB − sinA sinB. Cosine sum has a minus sign (opposite to the +).
sin(A − B) equals: — sinA cosB − cosA sinB.sin(A − B) = sinA cosB − cosA sinB.
cos(A − B) equals: — cosA cosB + sinA sinB.cos(A − B) = cosA cosB + sinA sinB. Cosine difference has a plus sign.
The memory trick for sum/difference identities: cosine formulas have __ to the sign: — The opposite sign.cos(A+B) has − even though we are adding, cos(A−B) has + even though we are subtracting — opposite to the operation sign.
Using sum/difference identities, which is NOT a standard formula? — sin(A+B) = sinA + sinB.sin(A+B) ≠ sinA + sinB. This is a common misconception — the correct formula requires the cross terms.
The formula sin(A+B) = sinAcosB + cosAsinB is verified when A = B = 0: — sin(0) = sin(0)cos(0) + cos(0)sin(0) → 0 = 0 ✓.sin(0) = 0. Right side: (0)(1) + (1)(0) = 0. Identity checks out.
The domain of arcsin(x) is: — [−1, 1].arcsin(x) is defined only for x ∈ [−1, 1], since sine values range from −1 to 1.
The range of arcsin(x) is: — [−π/2, π/2].arcsin outputs angles in [−π/2, π/2] — the principal range (Q4 and Q1).
The range of arccos(x) is: — [0, π].arccos outputs angles in [0, π] — the principal range (Q1 and Q2).
What is arcsin(1/2)? — π/6.sin(π/6) = 1/2, and π/6 is in [−π/2, π/2], so arcsin(1/2) = π/6.
What is arccos(0)? — π/2.cos(π/2) = 0, and π/2 is in [0, π], so arccos(0) = π/2.
What is arctan(−1)? — −π/4.tan(−π/4) = −1, and −π/4 is in (−π/2, π/2), so arctan(−1) = −π/4.
What is the domain of arctan(x)? — (−∞, ∞).arctan accepts any real number as input because tan has range (−∞, ∞).
arccos(1) = ? — 0.cos(0) = 1, and 0 ∈ [0, π]. So arccos(1) = 0.
arcsin(0) = ? — 0.sin(0) = 0, and 0 ∈ [−π/2, π/2]. So arcsin(0) = 0.