Ejercicios resueltos trigonometría

1. IDENTIDADES
TRIGONOMÉTRICAS
GRUPO NÚMERO 4

2. 19. Si: 1 + Tanx = U. Secx
1 – Tanx = V. Secx
Halla: 𝑈2 + 𝑉2
 1 + Tanx = U. Secx
1 +
𝑆𝑒𝑛𝑥
𝐶𝑜𝑠𝑥
= U .
1
𝐶𝑜𝑠𝑥
Cosx + Senx = U
 1 – Tanx = V. Secx
1 - Tanx = V. Secx
1 -
𝑆𝑒𝑛𝑥
𝐶𝑜𝑠𝑋
= V.
1
𝐶𝑜𝑠𝑥
Cosx-Senx = V
 𝑈2 = 𝐶𝑜𝑠2x + 2 Senx.Cosx + 𝑆𝑒𝑛2x
𝑼𝟐 = 1 + 2 Senx.Cosx
 𝑉2
= 𝐶𝑜𝑠2
x + 𝑆𝑒𝑛2
x – 2(Senx.Cosx)
𝑽𝟐 = 1 – 2 Senx.Cosx
 𝑼𝟐 + 𝑽𝟐 = 2
Tan = Sen/Cos
Cosx . Secx = 1
𝐶𝑜𝑠2
x + 𝑆𝑒𝑛2
x = 1

3. 20. Si: Cosx + 𝐶𝑜𝑠2
x = 1
Calcula: 𝑆𝑒𝑛2𝑥 +
𝑆𝑒𝑛4𝑥
Cosx = 1- 𝐶𝑜𝑠2x
Cosx = 𝑺𝒆𝒏𝟐𝒙
𝐂𝐨𝐬𝟐
x = 𝐒𝐞𝐧𝟒
𝐱
𝑺𝒆𝒏𝟐𝒙 + 𝑺𝒆𝒏𝟒𝒙
Cosx + 𝐶𝑜𝑠2x = 1
21. Reduce:
Q =
𝑇𝑔𝑥
𝑆𝑒𝑐𝑥 −𝐶𝑜𝑠𝑥
Q =
𝑆𝑒𝑛𝑥
𝐶𝑜𝑠𝑥
1 −𝐶𝑜𝑠2𝑥
𝐶𝑜𝑠𝑥
Q =
𝑆𝑒𝑛𝑥.𝐶𝑜𝑠𝑥
𝐶𝑜𝑠𝑥.𝑆𝑒𝑛2𝑥
Q =
1
𝑆𝑒𝑛𝑥
Q = Cscx
Senx
𝐶𝑜𝑠2
x + 𝑆𝑒𝑛2
x = 1
Tgx =
Senx
𝐶𝑜𝑠𝑥
(Senx)(Cscx) = 1

4. 22. El equivalente de:
E =
𝐶𝑜𝑠𝑥
1+𝑆𝑒𝑛𝑥
es:
E =
𝐶𝑜𝑠𝑥 ( 1 −𝑆𝑒𝑛𝑥 )
1+𝑆𝑒𝑛𝑥 ( 1 −𝑆𝑒𝑛𝑥 )
E =
𝐶𝑜𝑠𝑥 ( 1 −𝑆𝑒𝑛𝑥 )
1 − 𝑆𝑒𝑛2𝑥
E =
𝐶𝑜𝑠𝑥 ( 1 −𝑆𝑒𝑛𝑥 )
𝐶𝑜𝑠2𝑥
E =
1 −𝑆𝑒𝑛𝑥
𝐶𝑜𝑠𝑥
E =
1
𝐶𝑜𝑠𝑥
-
𝑆𝑒𝑛𝑥
𝐶𝑜𝑠𝑥
E = Secx - Tgx
23. El equivalente de:
M =
1+𝐶𝑜𝑠𝑥
1 −𝐶𝑜𝑠𝑥
M =
1+𝐶𝑜𝑠𝑥 (1+𝐶𝑜𝑠𝑥)
1 −𝐶𝑜𝑠𝑥 (1+𝐶𝑜𝑠𝑥)
M =
( 1+𝐶𝑜𝑠𝑥)2
𝑆𝑒𝑛2𝑥
M =
1+𝐶𝑜𝑠𝑥
𝑆𝑒𝑛𝑥
M = Cscx + Ctgx
- 𝐶𝑜𝑠2
x + 𝑆𝑒𝑛2
x = 1
- (Cosx)(Secx) = 1
- Tgx =
Senx
Cosx

5. 24. Simplifica:
K = 𝑆𝑒𝑛2x . 𝐶𝑜𝑠2x (𝑆𝑒𝑐2x + 𝐶𝑜𝑠2x)
K = 𝑆𝑒𝑛2x . 𝐶𝑜𝑠2x (
1
𝐶𝑜𝑠2𝑥
+
1
𝑆𝑒𝑛2𝑥
)
K = 𝑆𝑒𝑛2
x . 𝐶𝑜𝑠2
x (
𝑆𝑒𝑛2+ 𝐶𝑜𝑠2x
𝐶𝑜𝑠2x . 𝑆𝑒𝑛2x
)
K = 1

6. ¡GRACIAS!