Trigonometric Functions: questions with answers

Halving the angle halves the RANGE it lives in, so the half angle rarely stays where the whole angle was: 90° to 180° becomes 45° to 90°, which is the first quadrant. Reading the signs off the second quadrant instead answers for x rather than for x/2.

Minutes are sixtieths, not hundredths: 24′ is 24/60 = 2/5 of a degree, while writing 24 after a decimal point would make it 24/100 instead. The minus sign belongs to the whole measure, degrees and minutes together, so it cannot be attached to the 68 alone.

The angle has to be found before the arc: 40 minutes is 40/60 of a revolution, and a revolution is 2π radian, so θ = 88/21 radian. Using 40/60 itself as the angle forgets the 2π and shrinks the answer to 6 cm. Dividing by the radius rather than multiplying turns a length into the wrong kind of quantity altogether.

The radian measure is a RATIO of two lengths, which is why it carries no unit of its own: 15 ÷ 3 = 5. Multiplying instead of dividing makes the angle grow with the radius, and quoting 15 treats a length in centimetres as though it were an angle.

Minutes are sixtieths: 20′ is a third of a degree, so the measure is −31/3 degree and the conversion gives −31π/540. Reading 20′ as 0.20 of a degree makes it a fifth instead of a third and lands on −51π/900. Multiplying by 180/π rather than π/180 inverts the conversion and produces a number in the hundreds, −1860/π.

Only a whole revolution restores the point, so only a multiple of 360° leaves the sine unchanged: sin 380° = sin 20°. Adding 180° instead moves the point through 1 half turns, which flips the sign of the sine every time. A quarter turn changes the function itself, turning the sine into a cosine.

In the second quadrant the first coordinate is negative and the second positive, so cos x = −3/5 while sin x = 4/5. The definition never asked the angle to be acute — it asks only where the terminal side lands, and a point of the second quadrant is as good as one of the first.

A half turn KEEPS each function and at most changes its sign; it is the quarter turn that exchanges them. That is clearest in the tangent: both coordinates flip, the two sign changes cancel in the quotient, and tan (π + x) = tan x exactly — which is why the tangent's period is π and not 2π.

The names mislead, and the definitions settle it: cosec x = 1/sin x, so the co-name goes with the plain function and the plain name with the co-function. The reciprocal of the secant is the cosine, not the sine. Nor is it the cosecant: that is the CO-function, a different relationship altogether.

A negative sine does not make the cosine negative too: in the fourth quadrant the point lies to the RIGHT of the y-axis and below the x-axis, so the cosine is positive while the sine is not. cos x = 3/5. Carrying the given ratio's minus sign across gives −3/5, which describes a point in the third quadrant instead.

Dropping the minus sign first would reduce a different angle altogether: −765° comes to 315°, while 765° comes to 45°, and the two lie in different quadrants. The sign is part of the angle, and reducing is adding or subtracting whole revolutions — never removing a sign.

Sign follows from position: in the fourth quadrant cos θ is positive and sin θ is negative, so their quotient tan θ is negative. Two negatives make a positive, which is why the tangent survives the third quadrant where both sine and cosine fail. Only in the first quadrant are all six positive at once.

Substituting means putting each value in place and keeping the squares where they stand, so line 2 is 4(1/2)² + 3(1/2)² − 4(1)². Substituting √3/2 for sin(π/6) reads the cosine row at that angle instead. Dropping the squares while substituting does two things at once and loses the exponents for good.

The sign inside the cosine formula always flips: a sum of angles gives a difference of products, and a difference of angles gives a sum. Copying the sign straight across gives cos A cos B + sin A sin B, which is the expansion of the OTHER one of the pair. The expansion that mixes sines with cosines belongs to the sine, not the cosine.

Two readings settle it: the vertical extent and the starting value. A curve trapped between the marked levels can only be the sine or the cosine, and which of those it is depends on whether it starts at 0 or at 1. A curve that escapes upwards between its dashed lines is the tangent, and confusing the two kinds mistakes a bounded function for an unbounded one.