Trigonometry Review

Identities

100

sin²(x)+cos²(x)=

1

200

tan²(x)+1=

sec²(x)

300

cot²(x)+1=

csc²(x)

400

sec²(x)-1=

tan²(x)

500

sin²(x)-1=

-cos²(x)

Identity Proofs

100

True or False: tan(x)sin(x)+cos(x)=sec(x)

True

200

True or False: sin(x)-sin(x)cos²(x)=sin³(x)

True

300

(1-cos²(x))(1+tan²(x))=

tan²(x)

400

tan²(x) / (tan²(x)+1)=

sin²(x)

500

[sin³(x)+cos³(x)] / [sin(x)+cos(x)] + sin(x)cos(x)=

1

Trig Equations

100

Find all x on the interval [0, 360] for: sin²(x)-1=0

x=90, x=270

200

Find all x on the interval [0, 360] for: cos³(x)=cos(x)

x = 0 or x = 360

300

Find all x on the interval [0, 360] for: 4cos²(x)-3=0

x=30 or x=330

400

Find all x on the interval [0, 360] for: 2cos²(x)+cos(x)=0

x=90, x=120, x=270, x=240

500

Find all x on the interval [0, 360] for: 1+sin(x)=2cos²(x)

x=30, x=150, x=270

Word Problems

100

A Ferris wheel begins at 1 foot above ground and reaches a maximum height of 99 feet above ground. It takes 30 seconds for one full revolution. Find a trigonometric equation for a rider's height above ground at t seconds.

h(t)=49cos(12(t-15))+50 or h(t)=49sin(12(t-7.5))+50

200

A ferris wheel with radius 25m makes a full rotation every 36 seconds. The bottom of the ride is 1m above ground. A passenger is at height 51m when t=0. Find a trigonometric equation to model their height in terms of time t.

h(t)=25cos(10t)+26 or h(t)=25sin(10(t+9))+26

300

A mass suspended from a spring is pulled down 2 ft. from its resting position. The mass is released at time t=0 and allowed to oscillate. If the mass returns to the initial position after 1 second, find two trigonometric equations (one using sin, one using cos) describing its height in terms of time t.

h(t)=2cos(360(t-0.5)) and h(t)=2sin(360(t-0.25))

400

A tsunami is a fast-moving ocean wave cause by an underwater earthquake. The water first goes down from its normal level, then rises an equal distance above its normal level, and finally returns to its normal level with a period of about 15 minutes. Suppose a tsunami with an amplitude of 10m approaches a pier in Honolulu with a normal water level of 9m. Write a cosine and sine equation modelling the depth of the water in terms of time t.

d(t)=10cos(24(t+3.75))+9

d(t)=10sin(24(t+7.5))+9 OR d(t)=-10sin(24t)+9

500

The average depth of water at the end of a dock is 6 ft. The depth increases/decreases 2 ft with the tide. If there is a high tide at 4 AM (t=4) and the tide goes from low to high every 6 hours, find a cosine function representing the depth of the water as a function of time t.

Double: at what two times in the first cycle is the tide 5 ft deep?

d(t)=2cos(30(t-4))+6

8 AM and 12 PM