Find the exact value of tan 3π/4.
AvichalbhartiIt is basically a study of the properties of the triangle and trigonometric function and their application in various cases. It helps in finding the angles and missing sides of a triangle with the help of trigonometric ratios. Commonly used angles are 0°, 30°, 45°, 60°, and 90°. With the help of only these angles, find the value of all other trigonometric angles.
In this triangle, Given an acute angle θ,
- The sine of θ is written as sinθ and defined as the ratio sinθ = perpendicular/hypotenuse
- The cosine of θ is written as cosθ and defined as the ratio cosθ = base/hypotenuse
- The tangent of θ is written as tanθ and defined as the ratio tanθ = perpendicular/base = sinθ/cosθ
Note The reciprocals of sine, cosine, and tangents also have names: they are cosecant, secant, and cotangent.
- The cosecant of θ is written as cosecθ and defined as cosecθ = 1/sinθ
- The secant of θ is written as secθ and defined as secθ = 1/cosθ
- The cotangent of θ is written as cotθ and defined as cotθ = 1/tanθ
There are three Pythagorean Identities
- sin2 θ + cos2 θ = 1
- tan2 θ + 1 = sec2 θ
- cot2 θ + 1 = cosec2 θ
Lets look at the complementary angles in trigonometric ratios,
- sin(90 + θ) = cosθ
- cos(90 + θ) = -sinθ
- tan(90 + θ) = -cotθ
- cot(90 + θ) = -tanθ
- sec(90 + θ) = -cosecθ
- cosec(90 + θ) = secθ
Trigonometric Ratio Table
The trigonometric angles have a fixed value. Some of the important angles are used in mathematics. These fixed values are used in calculations. Let’s take a look at the table given below,
Angles |
0° |
30° |
45° |
60° |
90° |
Sin θ |
0 |
1/2 |
1/√2 |
√3/2 |
1 |
Cos θ |
1 |
√3/2 |
1/√2 |
1/2 |
0 |
Tan θ |
0 |
1/√3 |
1 |
√3 |
∞ |
Cosec θ |
∞ |
2 |
√2 |
2/√3 |
1 |
Sec θ |
1 |
2/√3 |
√2 |
2 |
∞ |
Cot θ |
∞ |
√3 |
1 |
1/√3 |
0 |
Find the exact value of tan 3π/4.
Solution:
We have to find the value of tan3π/4
tan (3π/4) = tan(π/2 + π/4) = -cot(π/4) [as tan(90 + θ) = -cotθ here θ = π/4]
cot(π/4) = 1
So, tan(3π/4)=-1
Alternate Way
tan(180 – θ) = -tanθ
So, tan(3pi/4) = -tan(pi/4) = -1
Similar Questions
Question 1: Find the value of tan(5π/6)
Solution:
⇒ tan(5π/6) = tan(π/2 + π/3) = -cot(π/3) [as tan(90 + θ) = -cotθ here θ = π/3]
so cot(π/3) = 1/√3
So, tan(5π/6) = -1/√3
Question 2: Find the value of tan(5π/4)
Solution:
⇒ tan(180 + θ) = tanθ [As tanθ is positive in the third quadrant]
S0, tan(5π/4) = tan(π + π/4) which is equal to tan(π/4)
tan(π/4) = 1
Hence, tan(5π/4) = 1
Question 3: What is the value of cot(5π/6)?
Solution:
⇒ We know that, cotθ = 1/tanθ
tan(5π/6) = -1/√3 [We have deduced earlier]
So, cot(5π/6) = -√3
Alternative way
⇒ cot(5π/6) = cot(π/2 + π/3) = -tan(π/3) [as cot(90 + θ) = -tanθ here θ = π/3]
So tan(π/3) = √3
Hence, cot(5π/6) = -√3
Last Updated :
03 Jan, 2024
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