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Hitunglah cos2 (θ+60∘)+ cos2 (60∘−θ)−sin2 θ

Pertanyaan

Hitunglah cos squared space left parenthesis theta plus 60 degree right parenthesis plus space cos squared space open parentheses 60 degree minus theta close parentheses minus sin squared space theta  

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

nilai cos squared space left parenthesis theta plus 60 degree right parenthesis plus space cos squared space open parentheses 60 degree minus theta close parentheses minus sin squared space theta adalah 1 half.

Pembahasan

Ingat kembali rumus perbandingan trigonometri untuk penjumlahan dan selisih dua sudut berikut.

cos space open parentheses straight A minus straight B close parentheses equals cos space straight A space cos space straight B plus sin space straight A space sin space straight B 

cos space open parentheses straight A plus straight B close parentheses equals cos space straight A space cos space straight B minus sin space straight A space sin space straight B 

Maka soal di atas dapat dibuktikan dengan perhitungan berikut.

begin mathsize 12px style cos squared space left parenthesis theta plus 60 degree right parenthesis plus space cos squared space open parentheses 60 degree minus theta close parentheses minus sin squared space theta equals open parentheses cos space theta space cos space 60 degree minus sin space theta space sin space 60 degree close parentheses plus open parentheses cos space 60 degree space cos space theta plus sin space 60 degree space sin space theta close parentheses minus sin squared space theta equals open parentheses 1 half cos space theta minus 1 half square root of 3 cos space theta close parentheses squared plus open parentheses 1 half cos space theta minus 1 half square root of 3 cos space theta close parentheses squared minus sin squared space theta equals 1 fourth cos squared theta minus 1 half square root of 3 sin theta space cos space theta plus 3 over 4 sin squared space theta plus 1 fourth cos squared space theta plus 1 half square root of 3 sin theta space cos space theta plus 3 over 4 sin squared space theta minus sin squared space theta equals 2 over 4 cos squared space theta plus 2 over 4 sin squared space theta equals 1 half open parentheses cos squared space theta plus sin squared space theta close parentheses equals 1 half open parentheses 1 close parentheses equals 1 half end style      

Dengan demikian, nilai cos squared space left parenthesis theta plus 60 degree right parenthesis plus space cos squared space open parentheses 60 degree minus theta close parentheses minus sin squared space theta adalah 1 half.

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Tentukan himpunan penyelesaian tiap persamaan berikut untuk 0∘≤x≤360∘. e. 2 cos (2x+30∘) cos (2x−60∘)=21​3​

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