8.3 Values at Standard Angles (Derivation & Logic)

Step 1

Substitute standard values:

Step 2

$\cos 60^\circ = 1/2$ , so $\cos^2 60^\circ = 1/4$ .

Step 3

$\sec 30^\circ = 2/\sqrt{3}$ , so $\sec^2 30^\circ = 4/3$ .

Step 4

$\tan 45^\circ = 1$ , so $\tan^2 45^\circ = 1$ .

Step 5

$\sin 30^\circ = 1/2$ , $\cos 30^\circ = \sqrt{3}/2$ .

Step 6

Numerator: $5(1/4) + 4(4/3) - 1 = 5/4 + 16/3 - 1$ .

Step 7

Denominator: $(1/2)^2 + (\sqrt{3}/2)^2 = 1/4 + 3/4 = 1$ (This is a standard identity).

Step 8

Solve Numerator: LCM of 4, 3 is 12. $\frac{15 + 64 - 12}{12} = \frac{67}{12}$ .

Final Answer

67/12

Operation

Angle

Unit

Standard Values:

sin 0 = 0, sin 30 = 0.5, sin 45 = 0.707, sin 60 = 0.866, sin 90 = 1

cos 0 = 1, cos 30 = 0.866, cos 45 = 0.707, cos 60 = 0.5, cos 90 = 0

Select an operation and enter values.