# Solve a Right Triangle Given an Angle and a Leg

– WE WANT TO SOLVE

THE RIGHT TRIANGLE GIVEN THE LENGTH OF SIDE B

IS EIGHT UNITS, AND THE MEASURE OF ANGLE B

IS 38 DEGREES. SO IF ANGLE B IS 38 DEGREES,

SIDE B IS OPPOSITE THIS ANGLE, TO THIS WOULD BE SIDE B. WE KNOW THIS LENGTH

IS EIGHT UNITS. SO IF THIS IS ANGLE A,

THEN THIS WOULD BE SIDE A. IF THIS IS ANGLE C,

THIS WOULD BE SIDE C. LET’S START BY FINDING

THE MEASURE OF ANGLE A. WE KNOW THE SUM OF THE INTERIOR

ANGLES OF A TRIANGLE MUST BE 180 DEGREES, BUT IT’S ALSO TRUE

THAT THE TWO ACUTE ANGLES MUST HAVE A SUM OF 90 DEGREES, WHICH MEANS THE MEASURE

OF ANGLE A MUST BE EQUAL

TO 90 DEGREES – 38 DEGREES, WHICH IS EQUAL TO 52 DEGREES. NOW, WE NEED TO FIND

THE LENGTH OF SIDE A AND THE LENGTH OF SIDE C. TO DO THIS WE’LL HAVE TO USE

A TRIG EQUATION. SO THE FIRST WE HAVE TO DECIDE WHICH ANGLE WE WANT TO USE

IN THE TRIG EQUATION. LET’S SAY THAT WE WANT

TO USE ANGLE A. WE KNOW WE ALSO HAVE TO USE

THE LENGTH OF SIDE B BECAUSE WE CAN ONLY HAVE

ONE UNKNOWN IN OUR EQUATION. SO IF WE WANT TO USE THIS ANGLE

AND THIS SIDE, LET’S SAY WE WANT TO FIND

THE LENGTH OF SIDE C. WELL, B IS THE ADJACENT SIDE

TO ANGLE A, AND C IS THE HYPOTENUSE. AND SINCE THE COSINE FUNCTION

INVOLVES THE ADJACENT SIDE AND THE HYPOTENUSE SIDE

OF A RIGHT TRIANGLE, WE KNOW WE HAVE TO USE

THE COSINE OF 52 DEGREES IN OUR EQUATION. SO OUR EQUATION WILL BE

THE COSINE OF 52 DEGREES MUST EQUATION THE RATIO OF THE

ADJACENT SIDE OF THE HYPOTENUSE OR EIGHT OVER C. NOW, TO CLEAR THIS FRACTION

WE’LL MULTIPLY BOTH SIDES BY C. SO LET’S SIMPLIFY OUT. SO WE’LL HAVE C x COSINE 52

DEGREES MUST EQUAL EIGHT. AND NOW WE’LL DIVIDE BOTH SIGNS

BY A COSINE 52 DEGREES. SO C=THIS QUOTIENT HERE. SO NOW WE’LL GO

TO THE CALCULATOR. ONE OF THE MOST IMPORTANT THINGS

TO DO IS RECOGNIZE THAT WE HAVE TO HAVE

OUR CALCULATOR IN DEGREE MODE. SO I’LL PRESS THE MODE KEY. NOTICE HOW IN MY THIRD ROW THE

DEGREE FEATURE IS HIGHLIGHTED. SO I AM IN THE CORRECT MODE. IF I GO BACK TO THE HOME SCREEN, AND NOW I WANT TO FIND

THIS QUOTIENT HERE. EIGHT DIVIDED

BY COSINE 52 DEGREES. IF WE ROUND TO THE NEAREST TENTH

THIS WOULD BE 13.0. NOW WE NEED TO FIND THE LENGTH

OF SIDE A. NOW THAT WE KNOW THE LENGTH OF

TWO SIDES OF THE RIGHT TRIANGLE, WE COULD USE

THE PYTHAGOREAN THEOREM, BUT I’M GOING TO GO AHEAD

AND USE ANOTHER TRIG EQUATION. SO LET’S GO AHEAD

AND USE ANGLE A AGAIN. WE KNOW WE’RE GOING TO USE SIDE

A BECAUSE THAT’S THE UNKNOWN. NOW, WE HAVE THE OPTION

OF USING SIDE C, WHICH IS APPROXIMATELY 13.0,

OR SIDE B, WHICH WE KNOW

IS EXACTLY EIGHT UNITS. I’M GOING TO USE SIDE B BECAUSE I KNOW

THIS IS THE EXACT VALUE. IF I USE INVENTED VALUE

OF SIDE C THERE WILL BE MORE OF AN ERROR. REFERENCING ANGLE A, NOTICE HOW THE SIDE

A IS THE OPPOSITE SIDE, AND SIDE B IS THE ADJACENT SIDE. THE TRIG FUNCTION VALUE

THAT USES THE OPPOSITE SIDE AND ADJACENT SIDE OF A RIGHT

TRIANGLE IS TANGENT WHICH MEANS NOW WE’LL HAVE THE

EQUATION TANGENT 52 DEGREES MUST EQUAL A DIVIDED BY EIGHT. NOTICE IN THIS EQUATION

WE CAN SOLVE FOR A IN ONE STEP. WE CAN MULTIPLY BOTH SIDES

OF THE EQUATION BY EIGHT TO DETERMINE THE VALUE OF A. NOTICE HERE

THIS SIMPLIFIES OUT. SO A IS APPROXIMATELY

EQUAL TO THIS PRODUCT HERE, 8 x TANGENT 52 DEGREES. SO 8 TANGENT 52 DEGREES. SO THE LENGTH OF SIDE

A TO THE NEAREST TENTH WOULD BE APPROXIMATELY

10.2 UNITS. OKAY.

I HOPE YOU FOUND THIS HELPFUL.

Clear examples

Thanks mate!

you helped me through highschool you should be proud because my teacher cant help at all

thnx bro you're really helping me get my grade up

You literally taught me Precal 1 and Precal 2.

Thank you! You made the solution very simple.

BReh… My teacher couldn't explain this. THANK YOU!!

Thank you this was amazing help

this helped a lot thanks 🙂