sin^2 28*+ sin^2 62* + tan^2 38*- cot^2 52*+ 1/4 sec^2 30* = ?
Make use of your "co" factor property.
that is:
sin 28 = cos 62, and tan38 = cot52
note that 28+62 = 90, 38+52=90
sec 30 you must know as part of your trig repertoire
To solve this equation, we need to use trigonometric identities and properties. Let's break it down step by step:
sin²(28°) + sin²(62°) + tan²(38°) - cot²(52°) + 1/4sec²(30°)
Step 1: Recall the Pythagorean identity for sine and cosine:
sin²(θ) + cos²(θ) = 1
Step 2: Notice that cot(θ) is the reciprocal of tan(θ). Using the Pythagorean identity, we can rewrite cot²(θ) in terms of tan(θ):
cot²(θ) = 1/tan²(θ)
Step 3: Recall the trigonometric identity for secant:
sec²(θ) = 1 + tan²(θ)
Using this identity, we can rewrite sec²(30°) in terms of tan(30°):
sec²(30°) = 1 + tan²(30°)
Step 4: Evaluate the trigonometric functions:
sin²(28°) = (sin(28°))²
sin²(62°) = (sin(62°))²
tan²(38°) = (tan(38°))²
cot²(52°) = 1/tan²(52°)
sec²(30°) = 1 + tan²(30°)
Step 5: Substitute the trigonometric values into the equation:
(sin(28°))² + (sin(62°))² + (tan(38°))² - 1/tan²(52°) + 1/4(1 + tan²(30°))
Step 6: Calculate the trigonometric values:
Using a calculator or trigonometric table, find the sin and tan values for the given angles.
Step 7: Substitute the values obtained in Step 6 into the equation and perform the arithmetic calculations to get the final answer.
Please note that without the actual specific values for the trigonometric functions of these angles, we cannot provide the exact numerical solution.