Example: Determine a Tangent Function Value Using a Half Angle Identity

– IN THIS EXAMPLE, WE WANT
TO DETERMINE THE EXACT VALUE OF TANGENT PI/8
USING A HALF ANGLE IDENTITY AND NOTICE THAT WE HAVE
THREE OPTIONS FOR THE HALF ANGLE IDENTITY
FOR TANGENT. IN THIS VIDEO I’M GOING
TO USE THIS LAST IDENTITY BECAUSE I THINK IT
HAS THE SIMPLEST DENOMINATOR AND I THINK IT’S USUALLY EASIER
TO APPLY THESE IDENTITIES IF OUR ANGLE IS IN DEGREES SO LETS START BY CONVERTING
PI/8 RADIANS IN TO DEGREES. SO WE’LL MULTIPLY IT BY 180
DEGREES DIVIDED BY PI. SO NOTICE HOW THE PI’S
SIMPLIFY OUT. 8 AND 180 HAVE A COMMON FACTOR
OF 4. THERE’S TWO FOURS IN 8
AND THERE’S 45 FOURS IN 180 SO THIS GIVES US 45 DEGREES
DIVIDED BY 2 OR 22.5 DEGREES. SO LET’S REWRITE
THIS AS TANGENT 22.5 DEGREES AND THEN WE’LL DOUBLE THE ANGLE
TO IDENTIFY THE VALUE OF “A” USED HERE AS WELL AS HERE
ON THE RIGHT SIDE WHICH WE ACTUALLY HAD HERE
BEFORE WE DIVIDED. THIS IS THE SAME AS TANGENT
45 DEGREES DIVIDED BY 2. SO “A” IS EQUAL TO 45 DEGREES. NOW WE’LL GO AHEAD
AND APPLY THIS IDENTITY. THIS FUNCTION VALUE IS EQUAL
TO 1 – COSINE 45 DEGREES   DIVIDED BY SINE 45 DEGREES
AND WE STARTED TO BE FAMILIAR WITH THIS FUNCTION VALUE
FROM THE 45-45-90 TRIANGLE WHERE THE TWO LEGS ARE 1 AND THE
HYPOTHENUSE IS SQUARE ROOT 2. SO THE SINE OF 45 DEGREES
IS EQUAL TO 1/SQUARE ROOT 2 WHICH RATIONALIZES
TO SQUARE ROOT 2/2 AND BECAUSE THIS IS AN ISOSCELES
RIGHT TRIANGLE THE COSINE OF 45 DEGREES IS THE
SAME 1 DIVIDED BY SQUARE ROOT 2 WHICH RATIONALIZES
TO SQUARE ROOT 2 DIVIDED BY 2. SO WE CAN REWRITE THIS AS
1 – SQUARE ROOT 2/2 DIVIDED BY SQUARE ROOT 2/2. LET’S START BY MULTIPLYING THE
NUMERATOR AND DENOMINATOR BY 2 TO CLEAR THE FRACTIONS IN
THE NUMERATOR AND DENOMINATOR. SO WHEN WE DO THIS OUR NUMERATOR IS GOING TO BE 2 – AGAIN YOU CAN
THINK OF THIS AS 2/1 IF YOU WANT
SO THESE TWO’S SIMPLIFY OUT SO WE HAVE 2 – THE SQUARE ROOT
OF 2 DIVIDED BY–THE SAME THING IN THE DENOMINATOR THESE
TWO’S SIMPLIFY OUT SO WE’RE LEFT
WITH SQUARE ROOT 2. SO THE LAST THING WE PROBABLY
WANT TO DO IS RATIONALIZE THE DENOMINATOR SO LET’S GO
AHEAD AND REWRITE THIS. WE HAVE 2 – SQUARE ROOT 2
ALL OVER SQUARE ROOT 2 AND NOW WE’LL MULTIPLY THE
DENOMINATOR BY SQUARE ROOT 2 AND THE NUMERATOR
BY SQUARE ROOT 2. SO WE’RE GOING TO HAVE 2 SQUARE
ROOT 2 AND THEN – 2 DIVIDED BY 2 AND THEN WE CAN REWRITE THIS AS
2 SQUARE ROOT 2 ALL OVER 2 – 2/2 WHICH SIMPLIFIES NICELY
TO SQUARE ROOT 2 – 1.   REMEMBER WHEN YOU’RE DIVIDING
BY A MONOMIAL OR 1 TERM WE CAN BREAK IT UP
INTO INDIVIDUAL FRACTIONS. SO THE EXACT FUNCTION VALUE
OF TANGENT PI/8 RADIANS IS EQUAL TO SQUARE ROOT 2 – 1.

3 thoughts on “Example: Determine a Tangent Function Value Using a Half Angle Identity”

1. Kelly Goosecock says:

Thanks, that helps break it all down.

2. Jamie G. says:

Thank you from the bottom of my heart. lol

3. Cassandra Vera Newman says:

You are the most awesome trig teacher on or off-line I've had yet 🙂
I hope you are still out there saving students!
xxoo