Pertanyaan

Hitung ekspresi berikut tanpa menggunakan kalkulator maupun tabel trigonometri. a. cotan [ cos − 1 ( − 2 1 ) + cos − 1 ( 0 ) + tan − 1 ( 3 1 ) ]

Hitung ekspresi berikut tanpa menggunakan kalkulator maupun tabel trigonometri.

a. $cotan [cos^{- 1} (- \frac{1}{2}) + cos^{- 1} (0) + tan^{- 1} (\frac{1}{3})]$

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Jawaban terverifikasi

Jawaban

.

$cotan space open square brackets cos to the power of negative 1 end exponent space open parentheses negative 1 half close parentheses plus cos to the power of negative 1 end exponent space open parentheses 0 close parentheses plus tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses close square brackets equals 1 third square root of 3$ .

Pembahasan

Ingat kembali nilai-nilai trigonometri pada sudut istimewa berikut: Diketahui ekspresi . Misalkan: - Untuk - Untuk - Untuk Maka: Jadi, .

Ingat kembali nilai-nilai trigonometri pada sudut istimewa berikut:

Diketahui ekspresi $cotan space open square brackets cos to the power of negative 1 end exponent space open parentheses negative 1 half close parentheses plus cos to the power of negative 1 end exponent space open parentheses 0 close parentheses plus tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses close square brackets$ . Misalkan:

- Untuk cos to the power of negative 1 end exponent space open parentheses negative 1 half close parentheses

- Untuk cos to the power of negative 1 end exponent space open parentheses 0 close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row beta equals cell cos to the power of negative 1 end exponent space open parentheses 0 close parentheses end cell row cell cos space beta end cell equals 0 row cell cos space beta end cell equals cell 90 degree end cell row beta equals cell 90 degree end cell end table

- Untuk $tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses$

$table attributes columnalign right center left columnspacing 0px end attributes row gamma equals cell tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses end cell row cell tan space gamma end cell equals cell fraction numerator 1 over denominator square root of 3 end fraction end cell row cell tan space gamma end cell equals cell tan space 30 degree end cell row gamma equals cell 30 degree end cell end table$

Maka:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell cotan space open square brackets cos to the power of negative 1 end exponent space open parentheses negative 1 half close parentheses plus cos to the power of negative 1 end exponent space open parentheses 0 close parentheses plus tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses close square brackets end cell equals cell cotan space open parentheses alpha plus beta plus gamma close parentheses end cell row blank equals cell cotan space open parentheses 120 degree plus 90 degree plus 30 degree close parentheses end cell row blank equals cell cotan space 240 degree end cell row blank equals cell fraction numerator 1 over denominator tan space 240 degree end fraction end cell row blank equals cell fraction numerator 1 over denominator tan space open parentheses 180 degree plus 60 degree close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator tan space 60 degree end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of 3 end fraction cross times fraction numerator square root of 3 over denominator square root of 3 end fraction end cell row blank equals cell 1 third square root of 3 end cell end table end style$

Jadi, $cotan space open square brackets cos to the power of negative 1 end exponent space open parentheses negative 1 half close parentheses plus cos to the power of negative 1 end exponent space open parentheses 0 close parentheses plus tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses close square brackets equals 1 third square root of 3$ .