Identidades trigonometricas del arco simple
- 2. 1-. Simplifique la expresión M= (secα + 𝑡𝑎𝑛𝛼)(𝑐𝑠𝑐𝛼 − 1)
Solución
M= (secα + 𝑡𝑎𝑛𝛼)(𝑐𝑠𝑐𝛼 − 1)
M= secα𝑐𝑠𝑐𝛼 − 𝑠𝑒𝑐𝛼 + 𝑡𝑎𝑛𝛼 𝑐𝑠𝑐𝛼 − 𝑡𝑎𝑛𝛼
𝑡𝑎𝑛𝛼 + 𝑐𝑜𝑡𝛼
𝐬𝐞𝐧𝜶
𝐜𝐨𝐬𝛂
𝟏
𝒔𝒆𝒏𝛂
M= tanα + 𝑐𝑜𝑡𝛼 − 𝑠𝑒𝑐𝛼 +
𝑠𝑒𝑛𝛼
𝑐𝑜𝑠𝛼
1
𝑠𝑒𝑛𝛼
− 𝑡𝑎𝑛𝛼
M= 𝑐𝑜𝑡𝛼 − 𝑠𝑒𝑐𝛼 +
1
𝑐𝑜𝑠𝛼
M= 𝑐𝑜𝑡𝛼 − 𝑠𝑒𝑐𝛼 + 𝑠𝑒𝑐𝛼
M= 𝑐𝑜𝑡𝛼
- 3. 2-. Simplifique la expresión E =
𝑡𝑎𝑛𝜃+1
𝑡𝑎𝑛𝜃−1
-
𝑠𝑒𝑐𝜃+𝑐𝑠𝑐𝜃
𝑠𝑒𝑐𝜃−𝑐𝑠𝑐𝜃
Solución
E =
𝑡𝑎𝑛𝜃+1
𝑡𝑎𝑛𝜃−1
-
𝑠𝑒𝑐𝜃+𝑐𝑠𝑐𝜃
𝑠𝑒𝑐𝜃−𝑐𝑠𝑐𝜃
E =
𝑠𝑒𝑛𝜃
𝑐𝑜𝑠𝜃
+1
𝑠𝑒𝑛𝜃
𝑐𝑜𝑠𝜃
−1
-
1
𝑐𝑜𝑠𝜃
+
1
𝑠𝑒𝑛𝜃
1
𝑐𝑜𝑠𝜃
−
1
𝑠𝑒𝑛𝜃
E =
𝑠𝑒𝑛𝜃+𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃
𝑠𝑒𝑛𝜃−𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃
-
𝑠𝑒𝑛𝜃+𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜃
𝑠𝑒𝑛𝜃−𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜃
E =
𝑠𝑒𝑛𝜃+𝑐𝑜𝑠𝜃
𝑠𝑒𝑛𝜃−𝑐𝑜𝑠𝜃
-
𝑠𝑒𝑛𝜃+𝑐𝑜𝑠𝜃
𝑠𝑒𝑛𝜃−𝑐𝑜𝑠𝜃
E =
𝑠𝑒𝑛𝜃 + 𝑐𝑜𝑠𝜃 − 𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃
𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃
E =
0
𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃
E = 0
- 4. 3-. Simplifique la expresión E =
(𝑠𝑒𝑛𝛼+𝑐𝑜𝑠𝛼)2−1
𝑡𝑎𝑛𝛼
+
1−(𝑠𝑒𝑛𝛼−𝑐𝑜𝑠𝛼)2
𝑐𝑜𝑡𝛼
Solución
E =
𝑠𝑒𝑛2 𝛼 + 2𝑠𝑒𝑛𝛼𝑐𝑜𝑠𝛼 + 𝑐𝑜𝑠2 𝛼 − 1
𝑡𝑎𝑛𝛼
+
1 − (sen2α − 2senαcosα + cos2α)
cotα
E =
1 + 2𝑠𝑒𝑛𝛼𝑐𝑜𝑠𝛼 − 1
𝑡𝑎𝑛𝛼
+
1 − (1 − 2𝑠𝑒𝑛𝛼𝑐𝑜𝑠𝛼)
𝑐𝑜𝑡𝛼
1 1
E =
2𝑠𝑒𝑛𝛼𝑐𝑜𝑠𝛼
𝑠𝑒𝑛𝛼
𝑐𝑜𝑠𝛼
+
1 − 1 + 2𝑠𝑒𝑛𝛼𝑐𝑜𝑠𝛼
𝑐𝑜𝑠𝛼
𝑠𝑒𝑛𝛼
E = 2𝑐𝑜𝑠2
𝛼 + 2𝑠𝑒𝑛2
𝛼
E = 2(𝑐𝑜𝑠2
𝛼 + 𝑠𝑒𝑛2
𝛼)
1
E = 2
- 5. 4-. simplifique la expresión N=
𝑠𝑒𝑛𝑥
1+𝑐𝑜𝑠𝑥
+
1−𝑐𝑜𝑠𝑥
𝑠𝑒𝑛𝑥
(𝑠𝑒𝑐𝑥 + 1)
Solución
N =
𝑠𝑒𝑛𝑥
1+𝑐𝑜𝑠𝑥
+
1−𝑐𝑜𝑠𝑥
𝑠𝑒𝑛𝑥
( 𝑠𝑒𝑐𝑥 + 1)
N =
𝑠𝑒𝑛2 𝑥+(1−𝑐𝑜𝑠𝑥)(1+𝑐𝑜𝑠𝑥)
𝑠𝑒𝑛𝑥(1+𝑐𝑜𝑠𝑥)
(
1
𝑐𝑜𝑠𝑥
+ 1)
N =
𝑠𝑒𝑛2 𝑥+1−𝑐𝑜𝑠2 𝑥
𝑠𝑒𝑛𝑥(1+𝑐𝑜𝑠𝑥)
(1+𝑐𝑜𝑠𝑥)
𝑐𝑜𝑠𝑥
1 - 𝒄𝒐𝒔 𝟐 𝜶
𝒔𝒆𝒏 𝟐 𝜶
N =
2𝑠𝑒𝑛2 𝑥
𝑠𝑒𝑛𝑥𝑐𝑜𝑠𝑥
N =
2 𝑠𝑒𝑛𝑥
𝑐𝑜𝑠𝑥
N = 2tanx
