Angles past 90 degrees

Word	What it means
Standard position	Vertex at the origin, initial arm along the positive x-axis, measured counterclockwise for a positive angle. Only the terminal arm moves.
Initial arm	The fixed side, on the positive x-axis.
Terminal arm	The side that rotates. Where it ends up is the whole description of the angle.
Quadrant	One of four regions of the plane, numbered I to IV counterclockwise starting from the upper right.
Reference angle	The acute angle between the terminal arm and the x-axis — never the y-axis. Always between 0° and 90°.
Terminal point P(x, y)	Any point on the terminal arm. Its x and y carry the signs; the distance r is always taken positive.
r	The distance from the origin to P, found by r = √(x² + y²). Never negative.
Exact value	A value written as a fraction or a radical rather than a decimal: sin 60° = √3/2, not 0.866.
Ambiguous case	A sine law situation with two sides and a non-included angle, where zero, one or two triangles may fit the data.

Finding a reference angle. Quadrant I: the angle itself. Quadrant II: 180° − θ. Quadrant III: θ − 180°. Quadrant IV: 360° − θ. So 150° has reference 30°, and so does 210°.

Signs, without the mnemonic. sin θ = y/r, cos θ = x/r, tan θ = y/x, and r is always positive. So each ratio simply takes the sign of the coordinates. In quadrant II, x is negative and y is positive: sine positive, cosine negative, tangent negative. The CAST rule is this worked out in advance.

From a point. P(−3, 4): r = √(9 + 16) = 5, so sin θ = 4/5, cos θ = −3/5, tan θ = −4/3.

Exact values. cos 120° has reference 60°, so its size is ½, and quadrant II makes cosine negative: −½. Getting the size right and the sign wrong is the most common way to lose this mark.

Solving over 0° ≤ θ ≤ 360°. sin θ = 0.5 gives a reference angle of 30°, and sine is positive in quadrants I and II, so θ = 30° and 150°. A calculator returns only the first — the second has to be reasoned out every time.

Sine law: a/sin A = b/sin B = c/sin C  ·  Cosine law: c² = a² + b² − 2ab cos C

Use the sine law when you have two angles and any side (AAS or ASA), or two sides and an angle opposite one of them (SSA).

Use the cosine law when you have two sides and the angle between them (SAS), or all three sides (SSS). In both of those there is no matched pair to start the sine law with.

Worked example, sine law. a = 8, ∠A = 40°, ∠B = 65°: b = 8 sin 65°/sin 40° ≈ 11.3. B is the bigger angle, so b should be the longer side — and it is.

Worked example, cosine law. a = 5, b = 7, ∠C = 60°: c² = 25 + 49 − 2(5)(7)(0.5) = 39, so c ≈ 6.2. Note that the cosine law reduces to Pythagoras when C = 90°, because cos 90° = 0.

Given a, b and ∠A, compute the height from the third vertex: h = b sin A. Then compare:

• a < h — the side cannot reach the base. No triangle.
• a = h — it just reaches, perpendicular. One right triangle.
• h < a < b — it reaches in two places. Two triangles.
• a ≥ b — it reaches past the pivot, so only one closes. One triangle.

Worked example. a = 6, b = 9, ∠A = 30°. Then h = 9 sin 30° = 4.5. Since 4.5 < 6 < 9, there are two triangles. The second angle is 180° minus the first, so if sin B gives B ≈ 48.6°, the other possibility is B ≈ 131.4° — and you must check that it still leaves room for A, which it does.

• Giving only the first solution when solving over 0° to 360°. Almost every equation has two.
• Measuring the reference angle to the y-axis. It is always to the x-axis.
• Ignoring the ambiguous case in an SSA problem. If the question gives two sides and a non-included angle, check the height.
• Leaving the calculator in radian mode. Everything will be wrong and nothing will look obviously wrong.