# Find the exact value of tan 3π/4.

It 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.

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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

**sin**^{2}θ + cos^{2}θ = 1**tan**^{2}θ + 1 = sec^{2}θ**cot**^{2}θ + 1 = cosec^{2}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 Waytan(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