# 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.

ty sir

@Mathispower4u It's very hard to find example problems to the particular section of math one works on. You always come through though. The problems you've put up on college algebra and trigonometry have help keep me afloat in these subjects. The problems are dead on what I need to see/hear. Thanks.

On the first example, How come you don't work with sin?