Trig – 04/20/23

Find the exact values of sin 2x, cos 2x, and tan 2x.

313 HW: p382 33-42, 45, 47, 49, 51, 59, 61 Honors: 89, 91

Today’s Lesson: Half-Angle Formulas

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csc x = 3, π

2< x < π

Trig/Pre-Calculus

You will:•Use the Half-Angle Formulas to rewrite and evaluate trig functions.

Today’s Lesson: Half-Angle Formulas

Half-Angle Formulas

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sinu2=±

1−cosu2

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cosu2=±

1+ cosu2

u

u

u

uu

cos1

sin

sin

cos1

2tan

+=

−=

€

The signs ofsinu

2 and cos

u

2 depend on the quadrant in which

u

2 lies.

Find the exact value of the sin, cos, and tan of 105o.

Always look at the angle on the unit circle!

105o

210o

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sinx

2= ±

1 − cos x

2

€

x

2=105°

€

x = 210°

€

sin105° =1− cos210°

2

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cos210° = −3

2

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sin105° =1 − − 3 /2( )

2

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=1+

3

22

In QII, sin is positive.

€

=

2

2+

3

22

€

=

2 + 3

22

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=2 + 3

2

⎛

⎝ ⎜

⎞

⎠ ⎟1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

=2 + 3

4

€

=2 + 3

4

€

=2 + 3

2

Find the exact value of the sin, cos, and tan of 105o.

Always look at the angle on the unit circle!

105o

210o

2

cos1

2cos

xx +±=

€

x

2=105°

€

x = 210°

2

210cos1105cos

°+−=°

€

cos210° = −3

2

( )2

2/31105cos

−+−=°

223

1−−=

In QII, cos is negative.

223

22 −

−=22

32−

−= ⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

2

1

2

32

4

32−−=

4

32−−=

2

32−−=

Find the exact value of the sin, cos, and tan of 105o.

Always look at the angle on the unit circle!

105o

210o

x

xx

sin

cos1

2tan

−=

€

x

2=105°

€

x = 210°

°°−

=°210sin

210cos1105tan 2

1210sin,

2

3210cos −=°−=°

21

23

1

105tan−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=°

2123

22

−

+=

1

32

−+

= 32−−=

Find the exact value of the sin, cos, and tan of 75o.

Always look at the angle on the unit circle!

75o150o

2

cos1

2sin

xx −±=

€

x

2= 75°

€

x =150°

2

150cos175sin

°−=°

€

cos150° = −3

2

( )2

2/3175sin

−−=°

223

1+=

In QI, sin is positive.

223

22 +

=22

32+

= ⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

2

1

2

32

4

32+=

4

32+=

2

32+=

Find the exact value of the sin, cos, and tan of 75o.

Always look at the angle on the unit circle!

75o150o

€

cosx

2= ±

1+ cos x

2

€

x

2= 75°

€

x =150°

€

cos75° =1+ cos150°

2

€

cos150° = −3

2

( )2

2/3175cos

−+=°

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=1−

3

22

In QI, cos is positive.

€

=

2

2−

3

22

€

=

2 − 3

22

€

=2 − 3

2

⎛

⎝ ⎜

⎞

⎠ ⎟1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

=2 − 3

4

€

=2 − 3

4

€

=2 − 3

2

Find the exact value of the sin, cos, and tan of 75o.

Always look at the angle on the unit circle!

75o150o

€

x

2= 75°

€

x =150°

x

x

sin

cos175tan

−=°

x

xx

sin

cos1

2tan

−=

2

1150sin,

2

3150cos =°−=°

21

23

1

75tan⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=°

21

23

22 +

= 32+=

Find the exact values of sin2u, cos2u, and tan2u using the double-angle formula.

Rewrite the expression using the power-reducing formula.

2 4sin cosx x