Examples: Simplify and Evaluate a Trig Expression Using a Double Angle Identity

– IN THIS VIDEO WE’LL USE DOUBLE
ANGLE IDENTITIES TO SIMPLIFY AND THEN EVALUATE
THE FOLLOWING EXPRESSIONS. SO THE FIRST EXPRESSION
IS COSINE SQUARED PI DIVIDED BY 12 – SINE SQUARED
OF PI DIVIDED BY 12 WHICH FITS THE FORM
OF THIS FIRST IDENTITY. NOTICE ON THE RIGHT SIDE WE HAVE COSINE SQUARED “A” – SINE
SQUARED “A”=COSINE OF 2A. SO FOR THE FIRST EXPRESSION,
“A” IS EQUAL TO PI/12. SO WE CAN REWRITE THIS AS COSINE
OF 2A OR COSINE OF 2 x PI/12 PUT THE 2/1 THIS SIMPLIFIES. SO WE HAVE A COSINE PI/6
WHICH IS A 30 DEGREE ANGLE. LETS GO AHEAD AND SKETCH
OUR REFERENCE TRIANGLE AND THE COSINE ON 30 DEGREES
IS SQUARE ROOT 3 DIVIDED BY 2.   FOR THE SECOND EXPRESSION, WE HAVE 2 COSINE OF PI/4 x
COSINE PI/4 WHICH FITS THIS DOUBLE ANGLE IDENTITY
FOR SINE. NOTICE WE HAVE 2 SINE “A” COSINE
“A” ON THE RIGHT SIDE WHICH IS EQUAL TO SINE OF 2A. SO “A” IS EQUAL TO PI/4. SO THIS IS EQUAL TO SINE
OF 2A OR 2 x PI/4. AGAIN, THIS SIMPLIFIES NICELY. SO THIS EQUALS SINE
OF PI/2 WHICH IS EQUAL TO 1 AND WE CAN VERIFY THIS IF WE
NEED TO ON THE UNIT CIRCLE. HERE’S PI/2 RADIANS
ON THE UNIT CIRCLE. SINE THETA IS EQUAL TO Y. SO SINE PI/2 IS EQUAL TO 1. IN OUR LAST EXAMPLE, WE HAVE 2
COSINE SQUARED PI/2 – 1 WHICH FITS THE FORM
OF THIS IDENTITY HERE WHERE “A”=PI/2 AND SO THIS RIGHT SIDE IS EQUAL
TO COSINE OF 2 x “A” SO WE HAVE COSINE OF 2 x PI/2
IS JUST GOING TO BE PI AND COSINE OF PI IS EQUAL TO -1 AND THEN AGAIN,
IF WE NEED TO WE CAN VERIFY THIS ON THE UNIT CIRCLE. HERE’S THE TERMINAL SIDE
OF PI RADIANS. COSINE THETA IS EQUAL TO X. SO WE HAVE A COSINE FUNCTION
VALUE OF -1.