If sinθ=5/13,determine each of the following without the use of a calculator
(Hint use a sketch)(θ<90°)
1.4.1 cos θ
1.4.2 tanθ
sinθ=5/13 = y/r
so sketch a right-angled triangle with hypotenuse 13, opposite side (the y)
equal to 5.
x^2 + y^2 = r^2
x^2 + 25 = 169
x^2 = 144
x = 12 , everything is positive since we are in quadrant I
cosθ = 12/13
tanθ = 5/12
perhaps you recognized the common 5-12-13 right-angled triangle.
To determine the values of cos θ and tan θ, we can use the Pythagorean identity and the definition of tangent.
From the hint given, "use a sketch," we can sketch a right triangle to represent the given information. Since sin θ = 5/13, we can label the side opposite θ as 5 and the hypotenuse as 13.
Using the Pythagorean identity, we can find the adjacent side (x) of the triangle:
sin^2 θ + cos^2 θ = 1
(5/13)^2 + cos^2 θ = 1
25/169 + cos^2 θ = 1
cos^2 θ = 1 - 25/169
cos^2 θ = 144/169
Taking the square root of both sides, we can find the value of cos θ:
cos θ = ± (sqrt(144) / sqrt(169))
cos θ = ± (12/13)
Since θ lies in the first quadrant (θ < 90°), cos θ is positive:
cos θ = 12/13 (Answer: 1.4.1)
To find tan θ, we can use the definition of tangent:
tan θ = sin θ / cos θ
tan θ = (5/13) / (12/13)
tan θ = 5/12 (Answer: 1.4.2)
Therefore, without using a calculator, we find that cos θ = 12/13 and tan θ = 5/12.
To determine the values of cos θ and tan θ without using a calculator, we can make use of the given information that sin θ is equal to 5/13.
Given that sin θ = 5/13, let's use the Pythagorean identity to find cos θ:
sin² θ + cos² θ = 1
Substituting the value of sin θ:
(5/13)² + cos² θ = 1
Simplifying the equation:
25/169 + cos² θ = 1
cos² θ = 1 - 25/169
cos² θ = (169 - 25)/169
cos² θ = 144/169
Taking the square root of both sides:
cos θ = ± √(144/169)
Since θ is in the first quadrant (θ < 90°), cos θ is positive:
cos θ = √(144/169)
Now, let's determine the value of tan θ:
tan θ = sin θ / cos θ
Substituting the values of sin θ and cos θ:
tan θ = (5/13) / √(144/169)
tan θ = (5/13) * (13/12)
tan θ = 5/12
Therefore, the values are:
1.4.1 cos θ = √(144/169)
1.4.2 tan θ = 5/12