Razones trigonométricas de ángulos notables explores key trigonometric principles essential for understanding right triangles. Authored by Prof. Edwin Ronald Cruz Ruiz, this resource is designed for secondary education students studying mathematics. It includes solved exercises on notable angles such as 30°, 45°, and 60°, providing a comprehensive overview of trigonometric ratios. The document also features practice problems and directed tasks to reinforce learning. Ideal for students preparing for exams or seeking to improve their mathematical skills in trigonometry.

Key Points

  • Explains trigonometric ratios for notable angles like 30°, 45°, and 60°.
  • Includes solved exercises to aid understanding of right triangles.
  • Offers directed practice problems for secondary education students.
  • Covers essential concepts for mastering trigonometric functions.
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Trigonometry
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Trigonometry
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I.E 10214 – LA RAMADA – SALAS Matemática 4º de Secundaria
Prof. Edwin Ronald Cruz Ruiz
RAZONES TRIGONOMÉTRICAS
DE ÁNGULOS NOTABLES
Ejercicios Resueltos
Son aquellos triángulos rectángulos donde
conociendo las medidas de sus ángulos
agudos, se puede saber la proporción
existente entre sus lados.
Como por ejemplo:
1. Triángulo Notable de 45º
2. Triángulo Notable de 30º y 60º
3. Triángulo Notables Aproximados
a) Triángulo de 37º y 53º
b) Triángulo de 16º y 74º
c) Triángulo de 8º y 82º
1. Calcular: E = Sen
2
30º + Tg37º
Solución:
Reemplazando valores:
1E
4
3
4
1
4
3
2
1
E
2
2. Evaluar:
csc30º
cos60º45ºsen
E
2
Solución:
Reemplazando:
2
1
2
2
1
4
2
2
2
1
2
2
2
E =
2
1
k
k
k
45º
45º
2k
2k
60º
60º
30º
30º
k
k
k
2k
3
k
60º
30º
k
8º
82º
7k
k25
I.E 10214 – LA RAMADA – SALAS Matemática 4º de Secundaria
Prof. Edwin Ronald Cruz Ruiz
Práctica dirigida Nº 01
º53sec3º45secº30tg6E
01. Calcular: E = (sen30º + cos60º)tg37º
a) 1 b) 2 c) 1/4
d) 3/4 e) 4/3
02. Calcular
º45sec.2º37cos.10
º60.3º30.
tgsen
F
a) 1 b) 1/2 c) -1/3
d) 2 e) 2/3
03. Calcular:
a) 3 b) 5 c) 7
d) 9 e) 11
04. Calcular: E = sec37º + ctg53º - 2sen30º
a) 0 b) 1 c) 2
d) 3 e) 4
05. Resolver:
5xsen53º - 2sec60º = xtg45º + sec
2
45º
a) 1 b) 2 c) 3
d) 1/2 e) 1/4
06. Indicar el valor de “x” en:
tg(2x - 5º) = sen
2
30º + sen
2
60º
a) 15º b) 20º c) 25º
d) 30º e) 35º
07. Determine el valor de “m” para que “x” sea
30º.
1m
1m
x2cos
a) 2 b) 3 c) 4
d) 5 e) 6
08. Sea:
2
9θ
CotSec6θ.Tg3θ
2
9θ
Csc.Cos6θ.en3θ
θF
S
Para evaluar: = 10º
a)
13
b)
6
/ 8 c) 15
d)
15
/ 7 e) 17
09. Del gráfico hallar: ctg
a) 1,6
b) 1,7
c) 0,4
d) 0,6
e) 1,4
10. Del gráfico, hallar Ctg
a)
5
4
b)
4
7
c)
5
2
d)
5
7
e) 1
11. Del gráfico calcular:
seny
senx
E
a)
5
24
b)
5
4
c)
5
2
d)
24
e) 1
x + 3
2x + 1
5x - 3
45º
x
y
53º
45º
53º
10
5
I.E 10214 – LA RAMADA – SALAS Matemática 4º de Secundaria
Prof. Edwin Ronald Cruz Ruiz
Tarea Nº 01
1. Calcular:
E = (sec
2
45º + tg45º) ctg37º - 2cos60º
a) 0 b) 1 c) 2
d) 3 e) 4
2. Calcular: “x”
3xsec53º - tg45º = sec60º(sec45º + sen45º)
csc30º
a) 1 b) 2 c) 3
d) 4 e) 5
3. Calcular: E = (tg60º + sec30º - sen60º)
sec60º
a) 25/12 b) 25/24 c) 49/12
d) 49/24 e) 7/18
4. Calcular:
45ºSen
Cos30ºSen37ºSec60ºTg30º
E
2
a)
5
3
b)
5
311
c)
5
33
d)
3
35
e)
5
32
5. Calcular:
2
º45
tg
a)
2
b)
12
c)
12
d)
21
e)
22
6. Hallar “x”.
Siendo:
Csc30º
1
45º
x
Csc
a) –1 b) –2 c) 1
d) 2 e) 3
7. Determine tg en el gráfico.
a)
3
b)
3
3
c)
2
3
d)
6
3
e)
2
33
8. De la figura calcular a/b
a) 1
b) 2
c) 5
d) 7
e) 8
9. Del gráfico hallar
x
y
a) 1
b) 2
c) 3
d) 4
e) 6
30º
37º
x
y
y
a + b
a - b
53º
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FAQs

What are the key trigonometric ratios for notable angles?
The key trigonometric ratios for notable angles include sine, cosine, and tangent values. For example, sin(30°) equals 1/2, cos(30°) equals √3/2, and tan(30°) equals 1/√3. Similarly, for 45°, sin(45°) and cos(45°) both equal √2/2, while tan(45°) equals 1. Understanding these ratios is crucial for solving problems involving right triangles and is foundational for further studies in trigonometry.

How does the document assist with solving trigonometric problems?
The document provides a series of solved exercises that demonstrate how to apply trigonometric ratios to solve problems. Each example is carefully worked out, showing the step-by-step process of calculating values for angles and sides in right triangles. Additionally, practice problems are included to reinforce the concepts learned, allowing students to test their understanding and improve their problem-solving skills.

What types of exercises are included in the document?
The document includes a variety of exercises, ranging from basic calculations of trigonometric ratios to more complex problems involving multiple steps. Students can find exercises that require evaluating expressions involving sine, cosine, and tangent, as well as word problems that apply these concepts in real-world scenarios. This diversity in exercises helps cater to different learning styles and levels of understanding.

Who is the intended audience for this trigonometry resource?
This resource is primarily aimed at secondary education students who are studying mathematics, particularly those focusing on trigonometry. It is suitable for students preparing for exams or those looking to strengthen their understanding of trigonometric concepts. Teachers may also find it useful as a supplementary material for classroom instruction.