Trigonometric Expressions & Equations

Fundamental Identities

100

cos (-u)

cos u

200

1/sin u

csc u

300

sin u/cos u

tan u

400

sin²u + cos²u

1

500

1 + cot²u

csc²u

Simplifying Trig Expressions

100

sin x sec x

tan x

200

cot x/csc x

cos x

300

cot u sin u + tan u cos u

cos u + sin u

400

sin²x sec²x - sin²x

sin²x tan²x

500

cos²x + cos²x tan²x

1

Verifying Trig Identities

100

sec y cos y = 1

(1/cos y) cos y 

1

200

tan t cot t = 1

(sin t/cos t)(cos t/sin t)

(sin t cos t)/(cos t sin t)

1

300

tan (pi/2 - u) tan u = 1

cot u tan u

(1/tan u) tan u

1

400

cot²x sec²x - cot²x = 1

cot²x (sec²x-1)

cot²x tan²x

(1/tan²x) tan²x

1

500

cos x + sin x tan x = sec x

cos x + sin x (sin x/cos x)

cos x + sin²x/cos x

(cos²x/cos x) + (sin²x/cos x)

(cos²x + sin²x)/cos x

1/cos x

sec x

Solving Trig Equations

100

Solve for x on the interval [0, 2pi)

2 sin x + 1 = 0

x = 7pi/6, 11pi/6

200

Solve for x on the interval [0, 2pi)

sec x - 2 = 0

x = pi/3, 5pi/3

300

Solve for x on the interval [0, 2pi)

sqrt(3) csc x - 2 = 0

x = pi/3, 2pi/3

400

Solve for x on the interval [0, 2pi)

2 cos²x + cos x - 1 = 0

x = pi/3, 5pi/3, pi

500

Solve for x on the interval [0, 2pi)

sec²x - 1 = 0

x = 0, pi

Potpourri

100

Factor

x² + 4x -12

(x + 6)(x - 2)

200

opposite/hypotenuse

sin x

300

For (x, y), cosine is the . . .

x - value

400

The cofunction of sine

Cosine

500

Factor

x⁴ - 16

(x² + 4)(x² - 4)