Lei dos Senos e Cossenos: Exercícios Resolvidos

Lei dos Cossenos

A lei dos cossenos é utilizada em problemas que envolvem triângulos não retângulos, ou seja, os triângulos que não possuem um ângulo de 90°. Uma vez que não possuem ângulo reto, as relações trigonométricas (seno, cosseno e tangente) não podem ser aplicadas, o que culmina na utilidade da lei dos cossenos.

Veja, abaixo, a lei dos cossenos utilizada para descobrir lados e ângulos:

a² = b² + c² – 2·b·c·cos a
b² = a² + c² – 2·a·c·cos b
c² = a² + b² – 2·a·b·cos c

Nas fórmulas acima temos os lados a, b e c, na qual o lado que desejamos descobrir ou seu valor ou seu ângulo deve vir do lado esquerdo da igualdade, logo antes do sinal de igual.

Perceba, nos exemplos abaixo, como resolver problemas utilizando a lei dos cossenos.

1) Descubra o valor do lado X do no triângulo abaixo.

x² = b² + c² – 2·b·c·cos x
x² = 3² + 4² – 2 * 3 * 4 * cos60°

Nesse momento, precisamos aplicar conhecimentos de trigonometria e saber que o cosseno de 60° vale ½.

x² = 9 + 16 – 24 * ½
x² = 25 - 12
x² = 13
x = 13

2) Calcule o valor do cosseno do ângulo x.

a² = b² + c² – 2·b·c·cos x
7² = 5² + 6² – 2 * 5 * 6 * cos x
49 = 25 + 36 – 60cos x
49 = 61 – 60cos x

-12 = -60cos x

= cos x

cos x = 1/5

Lei dos Senos

O fundamento matemático é denominado lei dos senos porque determina que a relação do seno de um determinado ângulo é sempre proporcional à medida do lado oposto a esse ângulo.

Esse teorema define que sempre será constante a relação entre o seno e a medida do seu lado dentro de um triângulo.

Para o mesmo triângulo ABC acima temos que:

Para compreender melhor, digamos que o ângulo a vale 60° e o ângulo b 45°. Desse modo, o ângulo c valerá 75°. A partir disso podemos realizar as seguintes relações:

Perceba, no exemplo abaixo, como resolver problemas utilizando a lei dos senos.

1) No triângulo a seguir, determine a medida do lado AC, tendo em vista as medidas presentes nele. (Use √2 = 1,4 e √3 = 1,7).

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" style="height:186px; width:191px" />

Sendo B = 45°, A = 60° e BC = 10:

X = 8,2

Exercícios resolvidos Lei dos Cossenos

A relação entre os três lados de um triângulo e o cosseno de um dos três ângulos determina o que chamamos de lei dos cossenos.

a2 = b2 + c2 – 2·b·c·cos a

Essa lei de fácil aplicação demanda conhecimentos básicos de trigonometria e o entendimento que o lado do triângulo antes do sinal da igualdade deve ser aquele que está do lado oposto ao ângulo abordado na equação.

Veja, abaixo, alguns exercícios que demandam a aplicação da lei dos cossenos:

1) (UF- Viçosa) Dois lados de um terreno de forma triangular medem 15 m e 10 m, formando um ângulo de 60°, conforme a figura abaixo:

O comprimento do muro necessário para cercar o terreno, em metros, é:

a) 5(5 + √15)
b) 5(5 + √5)
c) 5(5 + √13)
d) 5(5 + √11)
e) 5(5 + √7)

A questão pede para calcularmos o perímetro do triângulo acima, mas antes precisamos descobrir o valor do lado oposto ao ângulo de 60°, assim temos:

a² = b² + c² – 2·b·c·cos a
a² = 15² + 10² – 2 * 15 * 10 * cos 60°
a² = 225 + 100 – 300 * ½
a² = 325 – 150
a² = 175

a = √175

Fatorando temos que:

a = √5*5*7
a = 5√7

Como a questão pede o valor do perímetro desse triângulo, devemos somar os valores dos lados.

5√7 + 10 + 15
25 + 5√7
5*5 + 5√7
5(5+√7), letra e

2) (UF- Juiz de Fora) Dois lados de um triângulo medem 8 m e 10 m e formam um ângulo de 60°. O terceiro lado desse triângulo mede:

a) 2√21 m
b) 2√31 m
c) 2√41 m
d) 2√51 m
e) 2√61 m

Sabemos que entre os lados que medem 8 m e 10 m, existe um ângulo de 60°. Desse modo, esse ângulo é oposto ao terceiro lado que devemos descobrir.

a² = b² + c² – 2·b·c·cos a
a² = 8² + 10² – 2 * 8 * 10 *cos 60°
a² = 64 + 100 – 160 * ½
a² = 164 – 80
a² = 84

a = √84
a = √2*2*21
a = 2√21, gabarito letra a.

3) (UNESP-SP-2009) Paulo e Marta estão brincando de jogar dardos. O alvo é um disco circular de centro O. Paulo joga um dardo, que atinge o alvo num ponto, que vamos denotar por P; em seguida, Marta joga outro dardo, que atinge um ponto denotado por M, conforme figura.

Sabendo-se que a distância do ponto P ao centro O do alvo é PO = 10 cm, que a distância de P a M é = 14 cm e que o ângulo PÔM mede 120°, a distância, em centímetros, do ponto M ao centro O é

Chamando a de 14 cm, c de 10 cm, e a distância MO a ser descoberta de b, temos que:

a² = b² + c² – 2·b·c·cos a
14²= x² + 10² – 2 * x * 10 * cos 120°
196 = x² + 100 + 20x * (-1/2)
96 = x² + 10x
x² + 10x – 96 = 0

a = 1
b = 10
c = -96

Como um número negativo não serve, a distância MO é de 6 cm.

Exercícios Resolvidos Lei dos Senos

A equação que envolve a razão da medida de um lado pelo seno do seu ângulo oposto é chamada de lei dos senos. Demandando noções básicas de trigonometria, a lei dos senos nada mais é do que uma proporção entre os lados e senos dos ângulos opostos dentro de um triângulo.

Veja, abaixo, como aplicar o conceito de lei dos senos nos mais diversos concursos públicos do país.

1) (UFU-MG) Considere o triângulo retângulo a seguir.

Sabendo-se que α = 120°, AB = AC = 1 cm, então AD é igual a:

Observe que o enunciado informa que AB e AC são iguais, logo nesse triângulo isósceles os ângulos B e C valem 45°.

Focando no triângulo ADB, sabemos que o ângulo alfa vale 120° e é oposto ao lado AB e que o ângulo B é igual a 45° e oposto ao lado AD. Desse modo:

2) (Mackenzie – SP) Três ilhas A, B e C aparecem num mapa em escala 1:10000, como na figura. Das alternativas, a que melhor se aproxima de distância entre as ilhas A e B é:

a) 2,3 km
b) 2,1 km
c) 1,9 km
d) 1,4 km
e) 1,7 km

Primeiramente, se temos um triângulo cujos dois ângulos medem 105° e 30°, o terceiro medirá 45°.

Como a questão pede a medida do lado AB, deveremos relacioná-lo ao seu ângulo oposto, o ângulo de 45°. Veja:

Agora, convertendo o valor no desenho para o valor real temos: 16,97 * 10000 = 169700 cm.

Sendo assim, convertendo de centímetros para quilômetros teremos aproximadamente 1,7km.

3) (UFSM) Na instalação das lâmpadas da praça de alimentação, a equipe necessitou calcular corretamente a distância entre duas delas, colocadas nos vértices B e C do triângulo, segundo a figura. Assim, a distância “d” é?

Para aplicar a lei dos senos no triângulo acima precisamos apenas identificar qual ângulo é oposto a qual lado. O ângulo de 30° é oposto ao lado AC que mede 50 metros, enquanto o lado BC, que é a medida d que o enunciado nos pede para encontrarmos, é oposto ao ângulo de 135°. Sendo assim, temos: