Angle Identities

θsin θcos θtan θ 0010 –π/6–1/2√3/2–√3/3 –π/4–√2/2√2/2–1 –π/3–√3/21/2–√3 –π/2–10Undef –2π/3–√3/2–1/2√3 –3π/4–√2/2 1 –5π/6–1/2–√3/2√3/3 –π–π0–10 –7π/6½–√3/2–√3/3 –5π/4√2/2–√2/2–1 –4π/3√3/2–1/2–√3 –3π/210Undef –5π/3√3/21/2√3 –7π/4√2/2 1 –11π/61/2√3/2√3/3 –2π010 θsin θcos θtan θ 0010 π/61/2√3/2√3/3 π/4√2/2 1 π/3√3/21/2√3 π/210Undef 2π/3√3/2–1/2–√3 3π/4√2/2–√2/2–1 5π/61/2–√3/2–√3/3 π0–10 7π/6–1/2–√3/2√3/3 5π/4–√2/2 1 4π/3–√3/2–1/2√3 3π/2–10Undef 5π/3–√3/21/2–√3 7π/4–√2/2√2/2–1 11π/6–1/2√3/2–√3/3 2π2π010

We can find patterns in the trig functions One such pattern set is called Negative Angles Negative Angle identities are as follows sin (–θ) = –sin (θ) cos (–θ) = cos (θ) tan (–θ) = –tan (θ)

The values for each of these cofunctions are related to one another by a phase shift The cofunction identities are: sin (π/2 – θ) = cos (θ) cos (π/2 – θ) = sin (θ) tan (π/2 – θ) = cot (θ)

We can use the angle identities to prove relations Use the rules here to verify other trig identities Verify each identity: 1) csc (θ – π/2) = –sec θ 2) cot (π/2 – θ) = tan θ 3) tan (θ – π/2) = –cot θ

We can use the identities to find when different trig functions are equal Follow the same rules as verifying identities to simplify and then use the unit circle Solve each trig equation for 0 ≤ θ ≤ 2π 4) cos (π/2 – θ) = csc (θ) 5) tan 2 θ – sec 2 θ = cos (–θ)

For the next class complete #2 – 14 even on page 804.

Sum and Difference Identities

Angle Sums sin (A – B) = sin A cos B – cos A sin B cos (A – B) = cos A cos B + sin A sin B tan (A – B) = Angle Differences sin (A + B) = sin A cos B + cos A sin B cos (A + B) = cos A cos B – sin A sin B tan (A + B) = tan A + tan B 1 – tan A tan B tan A – tan B 1 + tan A tan B

Use the Sum and Difference Identities to quickly simplify 1) cos 50˚ cos 40˚ – sin 50˚ sin 40˚ cos 90˚ = 0 2) sin 100˚ cos 170˚ + cos 100˚ sin 170˚ sin 270 ˚ = -1 Find the exact value 3) cos 105˚ (√2 - √6)/4 4) sin 15˚ (√6 - √2)/4 5) tan 195˚ (√3/3 + 1)/(1 - √3/3)

Complete 18 – 36 even on page 804