Simplify the following trigonometric expressions: 7.1 ( 30^)/( 60^) + (360^ - theta) * 2theta. 7.2.1 Expand using the compound angle identity for sine.

Hey candy, good to see you again. Here are the solutions to your trigonometry problems. 7.1 Step 1: Evaluate the known trigonometric values and simplify the terms. We know that 30^ = (1)/(2) and 60^ = (1)/(2). Also, (360^ - ) = - (tangent is negative in the fourth quadrant). The expression becomes: ( 30^)/( 60^) + (360^ - ) · 2 = (1)/(2)(1)/(2) + (- ) · 2 = 1 - · 2 Step 2: Apply the double angle identity for 2 and express in terms of and . The identity for 2 is 2 . The identity for is ( )/( ). Substitute these into the expression: 1 - (( )/( )) · (2 ) Step 3: Simplify the expression. The terms cancel out: 1 - 2 = 1 - 2 ^2 Step 4: Recognize the final identity. The expression 1 - 2 ^2 is a double angle identity for 2. 2 = 1 - 2 ^2 Thus, the simplified expression is: 2 7.2.1 Step 1: Expand the terms using the compound angle identity for sine. The compound angle identity is (A ± B) = A B ± A B. For (60^ + 2x): (60^ + 2x) = 60^ 2x + 60^ 2x For (60^ - 2x): (60^ - 2x) = 60^ 2x - 60^ 2x Step 2: Add the two expanded expressions. (60^ + 2x) + (60^ - 2x) = ( 60^ 2x + 60^ 2x) + ( 60^ 2x - 60^ 2x) The terms 60^ 2x cancel out: = 2 60^ 2x Step 3: Substitute the value of 60^. We know that 60^ = sqrt(3)2. = 2 (sqrt(3)2) 2x = sqrt(3) 2x Step 4: Equate the simplified expression to k 2x and solve for k. Given that (60^ + 2x) + (60^ - 2x) = k 2x. From our simplification, we have: sqrt(3) 2x = k 2x By comparing the coefficients of 2x on both sides, we find: k = sqrt(3) The value of k is: sqrt(3) Send me the next one 📸