Math 102 Key Ideas. 1 Chapter 1: Triangle Trigonometry. 1. Consider the following right triangle: c b

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1 Math 10 Key Ideas 1 Chapter 1: Triangle Trigonometry 1. Consider the following right triangle: A c b B θ C a sin θ = b length of side opposite angle θ = c length of hypotenuse cosθ = a length of side adjacent to angle θ = c length of hypotenuse tan θ = b a = length of side opposite angle θ length of side adjacent to angle θ Remember: SOH CAH TOA. An angle in the coordinate plane is said to be in standard position if its vertex is at the origin and one of its sides (which we call the initial side) is on the positive x-axis. The other side is called its terminal side. y Terminal Side θ Initial Side x

2 3. Definition of trig functions on the plane: Consider r (x, y) θ x y sin θ = y length of side opposite angle θ = r length of hypotenuse cosθ = x length of side adjacent to angle θ = r length of hypotenuse tan θ = y x = length of side opposite angle θ length of side adjacent to angle θ = sin θ cosθ sec θ = 1 cos θ = r x csc θ = 1 sin θ = r r cot θ = 1 tan θ = x y = cosθ sin θ r = x + y 4. Law of Cosines: Helpful in the SAS and SSS cases. In any triangle ABC, with sides of length a, b, c: a = b + c bc cosa b = a + c ac cosb c = a + b ab cos C 5. Law of Sines: Helpful in the AAS and SSA (Warning! Ambiguous case!). In any triangle ABC, a sin A = b sin B = c sin C. 6. Area of a triangle: The area of a triangle containing an angle C with the sides a and b is 1 ab sin C. 7. Heron s Formula: The area of a triangle with sides a, b, c is s(s a)(s b)(s c), where s = 1 (a + b + c).

3 Chapter : Trigonometric Functions 1. Two angles which have the same terminal side are called coterminal. Coterminal angles are found by adding or subtracting integer multiples of π.. If you rotate your initial side in a counterclockwise direction, you get a positive angle. If you rotate in a clockwise direction, you get a negative angle. 3. The radian measure of an angle is defined to be the distance traveled along the unit circle by the point P as it moves from its starting position on the initial side to its final position on the terminal side of the angle. 4. Here is the unit circle, with radians and degrees: 5. Unit conversions To convert radians to degrees, multiply by 180 π. To convert degrees to radians, multiply by π A central angle of θ radians in a circle of radius r subtends an arc of length: s = θr.

4 7. Linear and angular speed: Distance traveled by P Linear speed = = s Time elapsed t = θr t Angle traced by OP Angular speed = = θ Time elapsed t Notice: Linear speed = Angular speed r. 8. For every real number t, 1 sin t 1 and 1 cost Notation: (cost) 3 is written cos 3 t sin t 3 means sin(t 3 ) 10. Pythagorean Identities: For every real number t, sin t + cos t = 1. For every number t in the domain of both functions, 1 + tan t = sec t. For every number t in the domain of both functions, 1 + cot t = csc t. 11. Periodicity Identities: For every real number t, For every real number t, sin(t ± π) = sin t. For every real number t, cos(t ± π) = cos t. For every number t in the domain, tan(t ± π) = tant. For every number t in the domain, sec(t + π) = sec t. For every number t in the domain, csc(t + π) = csc t. For every number t in the domain, cot(t + π) = cot t. 1. Negative Angle Identities: For every real number t, sin( t) = sin t. For every real number t, cos( t) = cos t. For every number t in the domain, tan( t) = tant. For every number t in the domain, sec( t) = sec t. For every number t in the domain, csc( t) = csc t. For every number t in the domain, cot( t) = cot t.

5 13. Graphs of the trig functions: Sine Cosine Tangent

6 Secant Cosecant Cotangent 14. If A 0 and b > 0, then both f(t) = A sin(bt + c) and g(t) = A cos(bt + c) have amplitude A period π b phase shift c b, where a wave of the graph begins at t = c b.

7 3 Chapter 3: Trigonometric Identities and Equations 1. Addition and Subtraction Identities: sin(x + y) = sin x cosy + cosxsin y sin(x y) = sin x cosy cosxsin y cos(x + y) = cosxcosy sin x sin y cos(x y) = cos x cosy + sin x sin y tan x + tany tan(x + y) = 1 tan x tan y tan x tan y tan(x y) = 1 + tan x tan y. Cofunction Identities: sin(π/ x) = cosx. cos(π/ x) = sin x. tan(π/ x) = cot x. cot(π/ x) = tan x. sec(π/ x) = csc x. csc(π/ x) = sec x. 3. Double-Angle Identities: sin x = sin x cosx cos x = cos x sin x = 1 sin x = cos x 1 tan x = tanx 1 tan x 4. Power-Reducing Identities: sin 1 cos x x = cos 1 + cos x x = tan 1 cos x x = 1 + cos x 5. Half-Angle Identities: sin x = ± 1 cosx 1 + cos x cos x = ± 1 cosx tan x = ± 1 + cos x = 1 cosx sin x = sin x 1 + cos x 6. Inverse Sine: For each ν with 1 ν, sin 1 ν = u where π/ u π/ exactly when sin u = ν.

8 7. Inverse Cosine: For each ν with 1 ν, cos 1 ν = u where 0 u π exactly when cosu = ν. 8. Inverse Tangent: For each real number ν, tan 1 ν = u where π/ < u < π/ exactly when tan u = ν. 4 Section 5.6: Polar Coordinates 1. Choose a point O in the plane (like the origin) and a half-line (called the polar axis) extending from O. A point P is given polar coordinates (r, θ), where r is the distance from P to O and θ is the angle with the polar axis as initial side and OP as terminal side. O θ r P(r, θ) Polar axis. If a point has polar coordinates (r, θ), then its rectangular coordinates (x, y) are x = r cosθ and y = r sin θ. 3. If a point has rectangular coordinates (x, y), then its polar coordinates (r, θ) satisfy r = x + y and tan θ = y x.