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Evaluate. cos −1 (sin 𝟓𝝅 𝟔 )

𝝅 𝟑

Given sin 𝜽 =𝒙, express cos 𝜽 as a function of 𝒙.

cos 𝜽 =± 𝟏− 𝒙 𝟐

Given csc 𝜽 = 𝟓 𝟒 , and 𝝅 𝟐 <𝜽<𝝅, find tan 𝜽 𝟐

tan 𝜽 𝟐 =𝟐 (Talk about bounds!)

Prove identity of cos 𝜽 cos 𝜽− sin 𝜽 = 𝟏 𝟏− tan 𝜽

𝟏 𝟏− tan 𝜽 = 𝟏 𝟏− sin 𝜽 cos 𝜽 = cos 𝜽 cos 𝜽− sin 𝜽

Simplify until one of the following: sin 𝜽 , cos 𝜽 , tan 𝜽, 𝟏 𝟏− 𝒄𝒐𝒔 𝟐 𝜽 𝟏+𝒔𝒊𝒏𝜽

𝟏− 𝒄𝒐𝒔 𝟐 𝜽 𝟏+𝒔𝒊𝒏𝜽 = 𝟏+𝒔𝒊𝒏𝜽− 𝒄𝒐𝒔 𝟐 𝜽 𝟏+𝒔𝒊𝒏𝜽 = 𝒔𝒊𝒏𝜽+ 𝒔𝒊𝒏 𝟐 𝜽 𝟏+𝒔𝒊𝒏𝜽 = 𝒔𝒊𝒏𝜽(𝟏+𝒔𝒊𝒏𝜽) 𝟏+𝒔𝒊𝒏𝜽) =𝒔𝒊𝒏𝜽

Range of the function, 𝒇 𝒙 = cos −𝟏 𝒙

𝟎, 𝝅 𝟐

If sin 𝜽= 𝟓 𝟖 , then cos 𝟐𝜽 =

± 𝟕 𝟑𝟐 *We need to know what quadrant 𝜽 is in to be able to find sin 𝟐𝜽

Explain why cos −𝑨 = cos 𝑨

Think of Unit Circle! Both angles A and –A are on the same side of the y-axis, so the x-value (cosine) remains the exact same!

Find amplitude, period, and phase shift of the sine wave.

Amplitude = 7, Period = 6, Phase Shift = 𝟒 𝝅

Write equation for the cosine graph below.

𝒚= 𝟏 𝟐 cos 𝑥 3 − 𝜋 4 +2

Evaluate. cos 𝟑𝟓° sin 𝟓𝟓° + sin 𝟑𝟓° cos 𝟓𝟓°

1

Use a right triangle ABC, to explain why sin 𝑨 = cos 𝑩 .

*This is why sine and cosine are considered cofunctions!

Using the formula 𝑨= 𝟏 𝟐 𝒃𝒉, find the area of the triangle .

𝟐𝟓 sin 𝟖𝟕° sin 𝟒𝟑° 𝟐 sin 𝟓𝟎°

Solve the triangle below.

𝑪=𝟏𝟖𝟎°−𝟒𝟎°−𝟒𝟓° 𝒃= 𝟒 sin 𝟒𝟓° sin 𝑪 𝒂= 𝟒 sin 𝟒𝟎° sin 𝑪

Solve the triangle below.

𝑩= sin −𝟏 ( 𝟑 sin 𝟒𝟎° 𝟒 ) 𝑨=𝟏𝟖𝟎°−𝟒𝟎°−𝑩 𝒂= 𝟒 sin 𝑨 sin 𝟒𝟎° 𝑩= sin −𝟏 ( 𝟑 sin 𝟒𝟎° 𝟒 ) 𝑨=𝟏𝟖𝟎°−𝟒𝟎°−𝑩 𝒂= 𝟒 sin 𝑨 sin 𝟒𝟎° *Need to find 2nd triangle if 𝟏𝟖𝟎°−𝑩+𝟒𝟎°<𝟏𝟖𝟎°. (Same steps for finding A and a, but now new B is 𝟏𝟖𝟎°−first B.)

Find the center and radius of the circle: 𝒙 𝟐 −𝟐𝒙+ 𝒚 𝟐 +𝟒𝒚=𝟒

Center is at (𝟏, −𝟐); Radius is 3

Find the vertices of the ellipse: 𝒙 𝟐 +𝟔𝒙+𝟗 𝒚 𝟐 −𝟏𝟖𝒚=−𝟗

(-6, 1) and (0, 1)

Write an equation for the ellipse:

(𝒙+𝟏) 𝟐 + (𝒚+𝟏 ) 𝟐 𝟒 =𝟏

Write an equation for the parabola:

(𝒚−𝟏) 𝟐 =−(𝒙−𝟐)

Find the coordinates of the vertex of the parabola: 𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+c

(− 𝒃 𝟐𝒂 , 𝒄− 𝒃 𝟐 𝟒𝒂 )

Solve the system below: 𝟐𝒙−𝒚=−𝟏 𝒙+ 𝟏 𝟐 𝒚= 𝟑 𝟐

( 𝟏 𝟐 ,𝟐)

Solve the system below: 𝟐𝒙+𝟒𝒚= 𝟐 𝟑 𝟑𝒙−𝟓𝒚=−𝟏𝟎

(− 𝟓 𝟑 ,𝟏)

Solve the system below: 𝒚= 𝒙 𝟐 −𝟒 𝒚=𝟔𝒙−𝟏𝟑

(3, 5)

Solve the system below: 𝒚=𝟐𝒙+𝟏 𝟐 𝒙 𝟐 + 𝒚 𝟐 =𝟏

𝟎,𝟏 and (− 𝟐 𝟑 ,− 𝟏 𝟑 )

Solve the system for x. 𝒂𝒙+𝒃𝒚=𝒑 𝒄𝒙+𝒅𝒚=𝒒

𝒙= 𝒒𝒃−𝒑𝒅 𝒃𝒄−𝒂𝒅

Express the sum using summation notation: 𝟐 𝟑 − 𝟒 𝟗 + 𝟖 𝟐𝟕 − …+ −𝟏 𝟏𝟐 ( 𝟐 𝟑 )

𝒌=𝟏 𝟏𝟏 (−𝟏 ) 𝒌+𝟏 ( 𝟐 𝟑 ) 𝒌

Find the first term and common difference of an arithmetic sequence if the 4th term is 3 and the 20th term is 35.

First Term = -3 Common Difference = 2

Find the sum: 𝑺=𝟐𝟏+𝟐𝟖+𝟑𝟓+ …+𝟑𝟎𝟏

6,601

Find the sum: 𝑺=𝟑+𝟏+ 𝟏 𝟑 + 𝟏 𝟗 +…

𝟗 𝟐

Find the sum: 𝒌=𝟏 𝟐𝟔 𝟑𝒌−𝟕

871

Final Jeopardy!!

Trig Identity Proofs

Simplify the equation ( cos 𝑨 − cos 𝑩 ) 𝟐 +( sin 𝑨 − sin 𝑩 ) 𝟐 =( cos 𝑨−𝑩 −𝟏 ) 𝟐 +( sin (𝑨−𝑩) ) 𝟐 to get one of the familiar sum or difference formulas!

( cos 𝑨 − cos 𝑩 ) 𝟐 +( sin 𝑨 − sin 𝑩 ) 𝟐 =( cos 𝑨−𝑩 −𝟏 ) 𝟐 +( sin (𝑨−𝑩) ) 𝟐 will simplify to cos 𝑨 cos 𝑩 + sin 𝑨 sin 𝑩 = cos (𝑨−𝑩)