Trigonometry and Analytic Geometry

Trigonometry and analytic geometry sit right beside algebra inside the 25% Mathematics group, and they pay off twice. The angles and identities you drill here reappear the moment you reach AC phasors, power factor, and three-phase relationships, and the coordinate methods sharpen the graph reading you need across the professional subjects. These items are also fast points: most are a single clean formula applied once. Master this lesson and you convert a whole cluster of the exam into near-automatic marks.

Angle measure: degrees and radians

Boards mix the two systems freely, so you must convert instantly. A full circle is 360 degrees or 2 pi radians, which gives the bridge 180 degrees = pi radians. To go from degrees to radians, multiply by pi/180; to go the other way, multiply by 180/pi.

Worked example: Convert 240 degrees to radians. Multiply by pi/180: 240(pi/180) = (240/180)pi = 4pi/3. As a check, 60 degrees is pi/3, and 240 is four of those, so 4 times pi/3 is 4pi/3. Going back, 5pi/6 radians is (5/6)(180) = 150 degrees.

The six functions and the identities

For an acute angle in a right triangle, the ratios are sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent. The other three are reciprocals: cosecant is 1/sine, secant is 1/cosine, and cotangent is 1/tangent. Memorizing the common Pythagorean triples (3-4-5, 5-12-13, 7-24-25, 8-15-17, 20-21-29) lets you read off exact ratios without a calculator.

The identities that score points most often are collected below.

Identity type	Statement
Pythagorean	sin^2(x) + cos^2(x) = 1; 1 + tan^2(x) = sec^2(x)
Sum	sin(A+B) = sin A cos B + cos A sin B; cos(A+B) = cos A cos B - sin A sin B
Difference	sin(A-B) = sin A cos B - cos A sin B; cos(A-B) = cos A cos B + sin A sin B
Double angle	sin(2x) = 2 sin x cos x; cos(2x) = 1 - 2 sin^2(x)

Worked example (Pythagorean): If sin(theta) = 7/25 for an acute theta, then cos(theta) = sqrt(1 - 49/625) = sqrt(576/625) = 24/25, which you could also read straight off the 7-24-25 triple. Worked example (double angle): With sin(theta) = 5/13 and cos(theta) = 12/13, sin(2 theta) = 2(5/13)(12/13) = 120/169. Worked example (sum): sin(75 degrees) = sin(45 + 30) = sin45 cos30 + cos45 sin30 = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6) + sqrt(2))/4.

Solving right and oblique triangles

A right triangle solves with the basic ratios and the Pythagorean theorem a^2 + b^2 = c^2. For any other (oblique) triangle you need two power tools. The law of sines, a/sin A = b/sin B = c/sin C, fits when you know an angle and its opposite side plus one more part. The law of cosines, c^2 = a^2 + b^2 - 2ab cos(C), fits when you know two sides and the included angle, or all three sides.

Worked example (law of cosines): Two sides of length 6 and 10 meet at a 120-degree angle. The third side is c = sqrt(6^2 + 10^2 - 2(6)(10)cos120) = sqrt(36 + 100 - 120(-0.5)) = sqrt(196) = 14. The trap is using cos120 = +0.5, which wrongly shrinks the answer to sqrt(76). Worked example (law of sines): In a triangle with angle A = 30 degrees, angle B = 45 degrees, and side a = 6 opposite A, side b = a sin B/sin A = 6(sin45/sin30) = 6(0.70711/0.5) = 6 sqrt(2), about 8.49.

Inverse trigonometric functions

The inverse functions answer "what angle gives this ratio". Write arcsin, arccos, and arctan, and remember their principal-value ranges so you pick the right quadrant: arcsin and arctan return values from -90 to 90 degrees, and arccos returns 0 to 180 degrees. For example, arctan(1) = 45 degrees because tan(45 degrees) = 1, and arcsin(0.5) = 30 degrees.

Analytic geometry: distance, midpoint, slope, and lines

Coordinate geometry turns pictures into arithmetic. For points (x1, y1) and (x2, y2):

Quantity	Formula
Distance	d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Midpoint	((x1 + x2)/2, (y1 + y2)/2)
Slope	m = (y2 - y1)/(x2 - x1)
Line (slope-intercept)	y = mx + b

Two lines are parallel when their slopes are equal and perpendicular when the slopes are negative reciprocals (their product is -1).

Worked example (distance): Between (-2, 3) and (4, -5), d = sqrt((4-(-2))^2 + (-5-3)^2) = sqrt(36 + 64) = sqrt(100) = 10. Worked example (line): The line through (-1, 4) and (2, -5) has slope (-5 - 4)/(2 - (-1)) = -9/3 = -3, and using point (2, -5): -5 = -3(2) + b gives b = 1, so y = -3x + 1.

Circles and the conic sections

A circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. When a circle is given in general form, complete the square on x and y to find its center and radius.

Worked example: For x^2 + y^2 - 6x + 8y - 11 = 0, group and complete the square: (x - 3)^2 + (y + 4)^2 = 11 + 9 + 16 = 36, so the center is (3, -4) and the radius is 6.

The other conic sections each carry a signature relationship.

Conic	Standard form	Key fact
Parabola	x^2 = 4py (opens up)	Focus at (0, p)
Ellipse	x^2/a^2 + y^2/b^2 = 1, a > b	Eccentricity e = sqrt(1 - b^2/a^2), between 0 and 1
Hyperbola	x^2/a^2 - y^2/b^2 = 1	Eccentricity e = sqrt(1 + b^2/a^2), greater than 1

Worked example (parabola): x^2 = 16y has 4p = 16, so p = 4 and the focus is (0, 4). Worked example (ellipse): With a = 13 and b = 5, e = sqrt(1 - 25/169) = sqrt(144/169) = 12/13. Worked example (hyperbola): With a = 5 and b = 12, e = sqrt(1 + 144/25) = sqrt(169/25) = 13/5.

Exam-day strategy

• Keep one conversion factor in your head, 180 degrees = pi radians, and derive every other angle from it instead of memorizing a long table.
• Reach for the law of sines when a side is paired with its opposite angle, and the law of cosines when you have two sides and the included angle or all three sides. Choosing the wrong law is the single most common oblique-triangle error.
• Watch the sign of the cosine for obtuse angles: cos(120 degrees) is negative, which makes the third side longer, not shorter.
• On circle items, always complete the square before reading off the center and radius, and remember the constant moves to the right side with a sign change.
• Separate the two eccentricity formulas by their sign: minus for the ellipse (e less than 1), plus for the hyperbola (e greater than 1). If your ellipse eccentricity comes out above 1, you used the wrong formula.
• For inverse trig, confirm the quadrant of your answer against the principal-value range before you commit, especially when the ratio is negative.