Solve frac{230^{circ}+cos^{2}45^{circ}-4tan^{2}30^circ}{2sec60^circ*sin30^circ+tan45^circ}

$$\frac { 230 ^ { \circ } + \cos ^ { 2 } 45 ^ { \circ } - 4 \tan ^ { 2 } 30 ^ { \circ } } { 2 \sec 60 ^ { \circ } \times \sin 30 ^ { \circ } + \tan 45 ^ { \circ } }$$

Evaluate

$$\frac{687500000000000000}{9928203230275509}\approx 69.247172329$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{230 + \cos^{2}(45) - 4 \tan^{2}(30)}{2 \cdot 1.1547005383792515 + \tan(45)}$$

Get the value of \cos(45) from trigonometric values table.

$$\frac{230+\left(\frac{\sqrt{2}}{2}\right)^{2}-4\left(\tan(30)\right)^{2}}{2\times 1.1547005383792515+\tan(45)}$$

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{230+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-4\left(\tan(30)\right)^{2}}{2\times 1.1547005383792515+\tan(45)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 230 times \frac{2^{2}}{2^{2}}.

$$\frac{\frac{230\times 2^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-4\left(\tan(30)\right)^{2}}{2\times 1.1547005383792515+\tan(45)}$$

Since \frac{230\times 2^{2}}{2^{2}} and \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-4\left(\tan(30)\right)^{2}}{2\times 1.1547005383792515+\tan(45)}$$

Get the value of \tan(30) from trigonometric values table.

$$\frac{\frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-4\times \left(\frac{\sqrt{3}}{3}\right)^{2}}{2\times 1.1547005383792515+\tan(45)}$$

To raise \frac{\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-4\times \frac{\left(\sqrt{3}\right)^{2}}{3^{2}}}{2\times 1.1547005383792515+\tan(45)}$$

Express 4\times \frac{\left(\sqrt{3}\right)^{2}}{3^{2}} as a single fraction.

$$\frac{\frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{4\left(\sqrt{3}\right)^{2}}{3^{2}}}{2\times 1.1547005383792515+\tan(45)}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{4\times 3}{3^{2}}}{2\times 1.1547005383792515+\tan(45)}$$

Multiply 4 and 3 to get 12.

$$\frac{\frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{12}{3^{2}}}{2\times 1.1547005383792515+\tan(45)}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{12}{9}}{2\times 1.1547005383792515+\tan(45)}$$

Reduce the fraction \frac{12}{9} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{4}{3}}{2\times 1.1547005383792515+\tan(45)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and 3 is 12. Multiply \frac{230\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}} times \frac{3}{3}. Multiply \frac{4}{3} times \frac{4}{4}.

$$\frac{\frac{3\left(230\times 2^{2}+\left(\sqrt{2}\right)^{2}\right)}{12}-\frac{4\times 4}{12}}{2\times 1.1547005383792515+\tan(45)}$$

Since \frac{3\left(230\times 2^{2}+\left(\sqrt{2}\right)^{2}\right)}{12} and \frac{4\times 4}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3\left(230\times 2^{2}+\left(\sqrt{2}\right)^{2}\right)-4\times 4}{12}}{2\times 1.1547005383792515+\tan(45)}$$

Multiply 2 and 1.1547005383792515 to get 2.309401076758503.

$$\frac{\frac{3\left(230\times 2^{2}+\left(\sqrt{2}\right)^{2}\right)-4\times 4}{12}}{2.309401076758503+\tan(45)}$$

Get the value of \tan(45) from trigonometric values table.

$$\frac{\frac{3\left(230\times 2^{2}+\left(\sqrt{2}\right)^{2}\right)-4\times 4}{12}}{2.309401076758503+1}$$

Add 2.309401076758503 and 1 to get 3.309401076758503.

$$\frac{\frac{3\left(230\times 2^{2}+\left(\sqrt{2}\right)^{2}\right)-4\times 4}{12}}{3.309401076758503}$$

Express \frac{\frac{3\left(230\times 2^{2}+\left(\sqrt{2}\right)^{2}\right)-4\times 4}{12}}{3.309401076758503} as a single fraction.

$$\frac{3\left(230\times 2^{2}+\left(\sqrt{2}\right)^{2}\right)-4\times 4}{12\times 3.309401076758503}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{3\left(230\times 4+\left(\sqrt{2}\right)^{2}\right)-4\times 4}{12\times 3.309401076758503}$$

Multiply 230 and 4 to get 920.

$$\frac{3\left(920+\left(\sqrt{2}\right)^{2}\right)-4\times 4}{12\times 3.309401076758503}$$

The square of \sqrt{2} is 2.

$$\frac{3\left(920+2\right)-4\times 4}{12\times 3.309401076758503}$$

Add 920 and 2 to get 922.

$$\frac{3\times 922-4\times 4}{12\times 3.309401076758503}$$

Multiply 3 and 922 to get 2766.

$$\frac{2766-4\times 4}{12\times 3.309401076758503}$$

Multiply -4 and 4 to get -16.

$$\frac{2766-16}{12\times 3.309401076758503}$$

Subtract 16 from 2766 to get 2750.

$$\frac{2750}{12\times 3.309401076758503}$$

Multiply 12 and 3.309401076758503 to get 39.712812921102036.

$$\frac{2750}{39.712812921102036}$$

Expand \frac{2750}{39.712812921102036} by multiplying both numerator and the denominator by 1000000000000000.

$$\frac{2750000000000000000}{39712812921102036}$$

Reduce the fraction \frac{2750000000000000000}{39712812921102036} to lowest terms by extracting and canceling out 4.

$$\frac{687500000000000000}{9928203230275509}$$