**Exact Values of Trigonometric Functions of Widely Used Angles : **

Let us list out the values of trigonometric functions at known angles.

(i) The values given above are all exact.

(ii) We observe that sin 30° and cos 60° are equal. Also sin 60° and cos 30° are equal.

(iii) The value of reciprocal ratios namely cosecant, secant and cotangent can be obtained using the above table.

(iv) The result cos 90° = 0 does not allow us to define tan 90° and sec 90°.

(v) Similarly sin 0° = 0 does not permit us to define csc 0° and cot 0°.

**Problem 1 : **

Find the value of csc 30°.

**Solution : **

We know that sin θ and csc θ are reciprocal to each other.

Then,

csc 30° = 1 / sin 30°

In the table above, sin 30° = 1/2.

Therefore,

csc 30° = 1 / (1/2)

csc 30° = 1 ⋅ (2/1)

csc 30° = (1 ⋅ 2) / 1

csc 30° = 2/1

csc 30° = 2

**Problem 2 : **

Find the value of sec 45°.

**Solution : **

We know that cos θ and sec θ are reciprocal to each other.

Then,

sec 45° = 1 / cos 45°

In the table above, cos 45° = 1/√2.

Therefore,

sec 45° = 1 / (1/√2)

sec 45° = 1 ⋅ (√2/1)

sec 45° = (1 ⋅ √2) / 1

sec 45° = √2/1

sec 45° = √2

**Problem 3 : **

Find the value of cot 60°.

**Solution : **

We know that tan θ and cot θ are reciprocal to each other.

Then,

cot 60° = 1 / tan 60°

In the table above, tan 60° = √3.

Therefore,

cot 60° = 1 / √3

**Problem 4 : **

Find the value of sec 90°.

**Solution : **

We know that cos θ and sec θ are reciprocal to each other.

Then,

sec 90° = 1 / cos 90°

In the table above, cos 90° = 0.

Therefore,

sec 90° = 1 / 0

sec 90° = not defined

**Problem 5 : **

Find the value of sin^{2}45° + cos^{2}45°.

**Solution : **

In the table above, sin 45° = 1/√2 and cos 45° = 1/√2.

Then,

sin^{2}45° + cos^{2}45° = (sin 45°)^{2} + (cos 45°)^{2}

sin^{2}45° + cos^{2}45° = (1/√2)^{2} + (1/√2)^{2}

sin^{2}45° + cos^{2}45° = 1/2 + 1/2

sin^{2}45° + cos^{2}45° = (1 + 1) / 2

sin^{2}45° + cos^{2}45° = 2/2

sin^{2}45° + cos^{2}45° = 1

**Problem 6 : **

Find the value of sin 30° ⋅ csc 30° + tan 30° ⋅ cot 60°.

**Solution : **

In the table above,

sin 30° = 1/2

csc 30° = 1 / sin 30° = 2

tan 30° = 1/√3

cot 60° = 1 / tan 60° = 1/√3

Then, the value of (sin 30° ⋅ csc 30° + tan 30° ⋅ cot 60°) is

= (1/2) ⋅ (2) + (1/√3) ⋅ (1/√3)

= 1 + 1/3

= 3/3 + 1/3

= (3 + 1) / 3

= 4/3

After having gone through the stuff given above, we hope that the students would have understood the exact values of trigonometric functions of widely used angles.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**