jumlahdan selisih dua sudut serta jumlah dan selisih sinus kosinus dan tangen materi yang perlu diingat a jumlah dan selisih dua sudut 1 sin a b sin a cos b cos a sin b 2 cos a b , misalnya untuk segitiga yang kecil nilai dari sin r 5 13 untuk segitiga yang besar juga sama aja nilai sin r 5 13 juga karena 25 65 itu juga sama dengan 5 13 terus biar Berdasarkanrumus aturan cosinus di atas, maka di dapatkan rumus untuk menghitung besar sudutnya : Supaya kamu lebih paham, kerjakan contoh soal di bawah ini yuk Squad! Segitiga ABC diketahui panjang sisi a = 5 cm, panjang sisi c = 6 cm dan besar sudut B = 60º. Rumustangen sudut ganda. Dengan menggunakan rumus sin a b untuk a b maka diperoleh. Cos a b cos a cos b sin a sin b rumus cosinus selisih dua sudut. Jawab cos 2x 1 2 cos 2x cos 60. Rumus sudut ganda untuk sin 1 2. Tan x tan α maka x α k 180. Rumus tangen jumlah dan selisih dua sudut. Ketika terdapat bentuk persamaan a cos 2 x b sin x cos x c Sedangkanpanjang BC dapat dihitung dengan rumus aturan cosinus karena diketahui satu panjang sisi dan besar dua sudut segitiga. Menghitung panjang BC: BC 2 = AC 2 + AB 2 ‒ 2 × AC × AC × cos A BC 2 = (5√2) 2 + (10√2) 2 ‒ 2 × 5√2 × 10√2 × ½ BC 2 = 50 + 200 ‒ 200 × ½ BC 2 = 50 + 200 ‒ 100 BC 2 = 150 BC = √150 = √ (25×6) = √25 × √6) = 5√6 cm 1 Siswa mampu membuktikan rumus aturan sinus dan aturan cosinus. B. Tugas Kelompok: Diskusikan dengan teman dalam kelompokmu masalah berikut dan buatlah kesimpulan tentang konsep aturan sinus dan aturan cosinus serta rumusnya. C. Masalah: ATURAN SINUS Pada segitiga ABC kita Tarik garis tinggi dari titik B ke AC, kita peroleh garis BD tegak lurus B IDENTITAS PERJUMLAHAN/ SELISIH SINUS DAN KOSINUS. Rumus jumlah dan selisih sinus dan kosinus merupakan bentuk manipulasi dari rumus hasil kali sinus dan kosinus yang telah dibahas sebelumnya. Rumus-rumusnya adalah sebagai berikut. sin A + sin B = + B). cos ½(A - B) sin A − sin B = 2.cos ½(A + B). sin ½(A - B) cos(A - B) = cos A cos B + sin A sin B Rumus cosinus selisih dua sudut: cos (A - B) = cos A cos B + sin A sin B. Untuk memahami penggunaan rumus cosinus jumlah dan selisih dua sudut, pelajarilah (A - B) = sin A cos B - cos A sin B = 3/5 . (-12/13) - (-4/5) . 5/13 = -36/65 + 20/65 = - 16/65. 3. Rumus Tangen Jumlah dan Untukmencari panjang BC dapat menggunakan rumus aturan sinus. Panjang BC adalah: b 2 = c 2 - 2ac cos B + a 2 (cos 2 B + sin 2 B) b 2 = c 2 + a 2 - 2ac cos B. Menggunakan analogi yang sama, kemudian diperoleh aturan cosinus untuk segitiga ABC sebagai berikut: a 2 = c 2 + b 2 - 2bc cos A. Theexpansion of sin(a - b) formula can be proved geometrically. To give the stepwise derivation of the formula for the sine trigonometric function of the difference of two angles geometrically, let us initially assume that 'a', 'b', and (a - b) are positive acute angles, such that (a > b).In general, sin(a - b) formula is true for any positive or negative value of a and b. TRIGONOMETRI1 PERBANDINGAN TRIGONOMETRI A Nilai Perbandingan Trigonometri Perhatikan segitiga berikut ! Y y r Sin = Cosec = r y x r r y Cos = Sec = r x y x X Tan = Cotan = O x x y Selanjutnya nilai perbandingan trigonometri suatu sudut segitiga dapat ditentukan dengan menggunakan daftar / tabel dan kalkulator. JNMb. Rumus-Rumus Trigonometri – Dulu kami pernah membuat postingan tentang rumus trigonometri SMA seperti trigonometri sudut ganda, selisih sudut, dan penjumlahan sudut. Kali ini kita akan belajar mengingat kembali apa itu trigonometri dan rumus aturan apa saja yang ada di dalamnya. Buat sebagian sobat hitung di rumah, trigonometri mungkin jadi materi dalam kategori susah dan ngga begitu disukai. Ah, kadang kita tida begitu serius PDKTnya, sehingga kita ngga begitu terasa rasa sukanya. Buat menambah PDKT kita tidak ada salahnya kita simak takjim sajian berikut. Apa itu Trigonometri Kalau sobat ditanya apa itu trigonometri kira-kira mau menjawab apa hayooo. Sobat, ternyata trigonometri berasal dari bahasa yunani “trigonon” yang bermakna segitiga dan “metron” yang berarti pengukuran. Trigonometri muncul di awal abad ke-3 masehi. Ia adalah salah satu cabang dari ilmu hitung matematika yang mempelajari segitiga meliputi semua aturan dalam penghitungan yang melibatkan sisi dan sudut dalam segitiga. Trigonometri terdiri dari sinus sin, cosinus cos, tangen tan, cotangen cot, secan sec, dan cosecan cosec. Untuk lebih memahami definisi trigonometri yuk simak gambar segitiga di bawah ini. Rumus Trigonometri Keterangan Sin α = b/c sisi depan dibagi sisi miring Cos α = a/c sisi samping dibagi sisi miring Tan α = b/a sisi depan dibagi sisi samping Cot α = a/b sisi samping dibagi sisi depan kebalikan dari tangen Sec α = c/a sisi miring dibagi sisi samping kebalikan dari cos Cosec α = c/b sisi miring dibagi sisi depan kebalikan dari sin Nilai Trigonometri Sudut-Sudut Istimewa Dalam trigonometri ada lima kaya poweranger sudut yang disebut sebagai sudut istimewa yaitu 0o, 30o, 45o, 60o, dan 60o. Penting bagi kita untuk mengetahui besarnya nilai trigonometri sudut-sudut tersebut karena rajin sekali muncul dalam soal ulangan atau ujian nasional. Rangkuman lengkap tentang nilai trigonometri dari sudut tersebut bisa di baca di tabel trigonometri sudut istimewa. Rumus-Rumus Identitas Trigonometri Nah ada istilah baru lagi ni, “identitas trigonometri”. Apa coba itu? Identitas trigonometri adalah sifat unik yang hanya dimiliki oleh trigonometri seperti sifat anomali pada air. Sifat itu hanya miliknya. Kalau dikelompokkan, sifat identitas ini bisa di bagi menjadi 3 kelas. Kelas yang pertama adalah identitas pebandingan, kelas kedua identitas kebalikan, dan yang terakhir identitas phytagoras. Berikur rumus trigonometri tersebut Relasi Sudut dalam Trigonometri Dalam trigonometri, ada relasi atar sudut-sudut. Sudut-sudut di kuadran II 90o-180o, kuadran III 180o-270o dan kuadran IV 270o-360o punya relasi dengan sudut-sudut di kuadran I 0o-90o. Berikut rumus-rumus sudut berelasi dalam trigonometri berikut trik untuk menghapalnya. 1. 180o – α –> Kuadran II sin 180o – α = sin α cos 180o – α = -cosα tan 180o – α = sin α 6. 90o – α –> Kuadran I sin 90o – α = cos α cos 90o – α = sin α tan 90o – α = cot α 2. 180o + α –> Kuadran III sin 180o + α = -sin α cos 180o + α = -cosα tan 180o + α = sin α 7. 90o + α –> Kuadran II sin 90o + α = cos α cos 90o + α = -sin α tan 90o + α = -cot α 3. 360o – α –> Kuadran IV sin 360o – α = -sin α cos 360o – α = cosα tan 360o – α = -sin α 8. 270o – α –> Kuadran III sin 270o – α = -cos α cos 270o – α = -sin α tan 270o – α = cot α 4. 360o + α –> Kuadran I sin 360o + α = sin α cos 360o + α = cosα tan 360o + α = sin α 9. 270o + α –> Kuadran IV sin 270o + α = -cos α cos 270o + α = sin α tan 270o + α = -cot α 5. untuk sudut -α –> Kuadran IV sin -α = -sin α cos -α = cosα tan -α = -sin α Rumus Cepat Rumus Cepat Pola lihat di kanan tanda = Sin → SinCos → CosTan → Tan Pola lihat di kanan tanda = Sin → CosCos → SinTan → Cot Penentuan +/- dilihat dari Kuadran, aturannya yang POSITIFKuadran I = All semuaKuadran II = hanya SIN Kuadran III = hanya TAN Kuadran IV = hanya COS sobat bisa mengingatnya ALL SIN TAN COS Jadi yang perlu sobat lakukan adalah menghafal pola dari sudut istimewa yang kelipatan 180o dan 90o kemudian tentukan hasilnya apakah positif atau negatif dengan menggunkan aturan ALL SIN TAN COS. Contoh soalnya seperti berikut Sobat ditanya berapa nilai sin 120o? sobat dapat menggunakan trik rumus trigonometri di atas. Cara I ingat, 120 = 90 + 30, jadi sin 120o dapat dihitung dengan Sin 120o = Sin 90o + 30o = Cos 30o nilainya positif karena soalnya adalah sin 120o, di kuadran 2, maka hasilnya positif Cos 30o = ½ √3 Cara II sobat bisa juga menggunakan rumus lain untuk soal trigonometri tersebut, 120o nilanya juga sama seperti 180o-80o. Sin 120o = Sin 180o – 60o = sin 60o = ½ √3 sama kan sobat hasilnya, hehehe 😀 Demikian sobat sajian kami tentang rumus trigonometri. Semoga bermanfaat. Untuk materi trigonometeri yang lain seperti grafik dan fungsi trigonometri dan pengukuran sudut akan kita sambung di postingan berikutnya. Selamat belajar. Buat orang tuamu bangga… 😀 Sin a cos b is an important trigonometric identity that is used to solve complicated problems in trigonometry. Sin a cos b is used to obtain the product of the sine function of angle a and cosine function of angle b. It can be obtained from angle sum and angle difference identities of the sine function. sin a cos b formula is written as 1/2[sina+b + sina-b]. In this article, we will explore the sin a cos b formula, its proof, and learn its application to solve various trigonometric problems with the help of solved examples. 1. What is Sin a Cos b Identity? 2. Proof of Sin a Cos b Formula 3. Application of Sin a Cos b Identity 4. FAQs on Sin a Cos b What is Sin a Cos b Identity? Sin a cos b is a trigonometric identity used to solve various problems in trigonometry. Sin a cos b is equal to half the sum of sine of the sum of angles a and b, and sine of difference of angles a and b. Mathematically, it is written as sin a cos b = 1/2[sina + b + sina - b], that is, it can be derived using the trigonometric identities sin a + b and sina - b. sin a cos b formula can be applied when the sum and difference of angles a and b are known, or when two angles a and b are known. Sin a Cos b Formula The formula for sin a cos b is given by, sin a cos b = 1/2[sina + b + sina - b]. The formula for sin a cos b can be applied when the compound angles a + b and a - b are known, or when values of angles a and b are known. Proof of Sin a Cos b Formula Now that we know the formula of sin a cos b, which is sin a cos b = 1/2[sina + b + sina - b], we will derive this formula using the trigonometric formulas and identities. Sin a cos b formula can be derived using the angle sum and angle difference formulas of the sine function. We will use the following trigonometric formulas sin a + b = sin a cos b + cos a sin b - 1 sin a - b = sin a cos b - cos a sin b - 2 Adding equations 1 and 2, we have sin a + b + sin a - b = sin a cos b + cos a sin b + sin a cos b - cos a sin b From 1 and 2 ⇒ sin a + b + sin a - b = sin a cos b + cos a sin b + sin a cos b - cos a sin b ⇒ sin a + b + sin a - b = sin a cos b + sin a cos b + cos a sin b - cos a sin b ⇒ sin a + b + sin a - b = 2 sin a cos b + 0 ⇒ sin a + b + sin a - b = 2 sin a cos b ⇒ sin a cos b = 1/2 [sin a + b + sin a - b] Hence, we have obtained the sin a cos b formula using the sin a + b and sin a - b identities. Application of Sin a Cos b Identity Since we have derived the sin a cos b formula, now we will learn how to apply the formula to solve simple trigonometric and integration problems. We will consider some examples based on sin a cos b identity and solve them step-wise. Let us understand the application of the sin a cos b formula by following the given steps Example 1 Express the trigonometric function sin 7x cos 3x as a sum of the sine function. Step 1 We will use the sin a cos b formula sin a cos b = 1/2 [sin a + b + sin a - b]. Identify the values of a and b in the formula. We have sin 7x cos 3x, here a = 7x, b = 3x. Step 2 Substitute the values of a and b in the formula sin a cos b = 1/2 [sin a + b + sin a - b] sin 7x cos 3x = 1/2 [sin 7x + 3x + sin 7x - 3x] ⇒ sin 7x cos 3x = 1/2 [sin 10x + sin 4x] ⇒ sin 7x cos 3x = 1/2 sin 10x + 1/2 sin 4x Hence, we can write sin 7x cos 3x as 1/2 sin 10x + 1/2 sin 4x as a sum of sine function. Example 2 Evaluate the integral ∫sin 2x cos 4x dx using the sin a cos b formula. Step 1 First, we will express sin 2x cos 4x as a sum of sine function using the formula sin a cos b = sin a cos b = 1/2 [sin a + b + sin a - b]. Identify a and b in sin 2x cos 4x. We have a = 2x, b = 4x. Step 2 Substitute the values of a and b in the formula sin a cos b = 1/2 [sin a + b + sin a - b] sin 2x cos 4x = 1/2 [sin 2x + 4x + sin 2x - 4x] ⇒ sin 2x cos 4x = 1/2 [sin 6x + sin -2x] ⇒ sin 2x cos 4x = 1/2 sin 6x - 1/2 sin 2x [Because sin-a = -sin a] Step 3 Substitute sin 2x cos 4x = 1/2 sin 6x - 1/2 sin 2x into the integral ∫sin 2x cos 4x dx. ∫sin 2x cos 4x dx = ∫ [1/2 sin 6x - 1/2 sin 2x] dx ⇒ ∫sin 2x cos 4x dx = 1/2 ∫sin6x dx - 1/2 ∫sin2x dx ⇒ ∫sin 2x cos 4x dx = 1/2[-cos6x]/6 - 1/2[-cos2x]/2 + C ⇒ ∫sin 2x cos 4x dx = -1/12 cos 6x + 1/4 cos 2x + C Hence, we have solved the integral ∫sin 2x cos 4x dx using sin a cos b formula and is equal to -1/12 cos 6x + 1/4 cos 2x + C. Important Notes on Sin a Cos b sin a cos b = 1/2[sina+b + sina-b] sin a cos b formula is applied when angles a and b are known, or when the sum and difference of angles a and b are known. sin a cos b formula is used to solve simple and complex trigonometric problems. Sin a cos b is equal to half the sum of sine of the sum of angles a and b, and sine of difference of angles a and b. Related Topics on Sin a Cos b sin a sin b cos a cos b sin of 2 pi cos 2x FAQs on Sin a Cos b What is Sin a Cos b in Trigonometry? Sin a cos b is an important trigonometric identity that is used to solve complicated problems in trigonometry given by sin a cos b = 1/2 [sin a + b + sin a - b] What is the Formula of Sin a Cos b? The formula of sin a cos b is sin a cos b = 1/2 [sin a + b + sin a - b] What is the Formula of 2 sin a cos b? The formula for 2 sin a cos b is given by, 2 sin a cos b = sin a + b + sin a - b Find the Exact Value of sin a cos b when a = 90° and b = 180°. Substitute a = 90° and b = 180° in sin a cos b = 1/2 [sin a + b + sin a - b]. sin 90° cos 180° = 1/2 [sin 90° + 180° + sin 90° - 180°] = 1/2 [sin 270° + sin-90°] = 1/2-1-1 = -1. Hence, sin a cos b = -1 when a = 90° and b = 180° How to Find sin a cos b formula? Sin a Cos b formula can be calculated using sina + b and sin a - b trigonometric identities. When is sin a cos b equal to 1/2 sin 2a? sin a cos b is equal to 1/2 sin 2a when a = b. When a = b in sin a cos b = 1/2 [sin a + b + sin a - b], we have sin a cos b = 1/2 [sin a + a + sin a - a] = 1/2 [sin 2a + 0] = 1/2 sin 2a. How to Prove sin a cos b Identity? Sin a cos b formula can be proved using the angle sum and angle difference formulas of the sine function. What is the Expansion of Sin a Cos b? The expansion of sin a cos b is given by sin a cos b = 1/2 [sin a + b + sin a - b]. What is the Difference Between Sin a Cos b Formula and Cos a Sin b Formula? Sin a cos b formula is the sum of sin a + b and sin a - b trigonometric identities, whereas cos a sin b formula is the difference of sin a + b and sin a - b trigonometric identities, that is, sin a cos b = 1/2 [sin a + b + sin a - b] and cos a sin b = 1/2 [sin a + b - sin a - b]. Sina Sinb is an important formula in trigonometry that is used to simplify various problems in trigonometry. Sina Sinb formula can be derived using addition and subtraction formulas of the cosine function. It is used to find the product of the sine function for angles a and b. The result of sina sinb formula is given as 1/2[cosa - b - cosa + b]. Let us understand the sin a sin b formula and its derivation in detail in the following sections along with its application in solving various mathematical problems. 1. What is Sina Sinb in Trigonometry? 2. Sina Sinb Formula 3. Proof of Sina Sinb Formula 4. How to Apply Sina Sinb Formula? 5. FAQs on Sina Sinb What is Sina Sinb in Trigonometry? Sina Sinb is the trigonometry identity for two different angles whose sum and difference are known. It is applied when either the two angles a and b are known or when the sum and difference of angles are known. It can be derived using angle sum and difference identities of the cosine function cos a + b and cos a - b trigonometry identities which are some of the important trigonometric identities. Sina Sinb formula is used to determine the product of sine function for angles a and b separately. The sina sinb formula is half the difference of the cosines of the difference and sum of the angles a and b, that is, sina sinb = 1/2[cosa - b - cosa + b]. Sina Sinb Formula The sina sinb product to difference formula in trigonometry for angles a and b is given as, sina sinb = 1/2[cosa - b - cosa + b]. Here, a and b are angles, and a + b and a - b are their compound angles. Sina Sinb formula is used when either angles a and b are given or their sum and difference are given. Proof of Sina Sinb Formula Now, that we know the sina sinb formula, we will now derive the formula using angle sum and difference identities of the cosine function. The trigonometric identities which we will use to derive the sin a sin b formula are cos a + b = cos a cos b - sin a sin b - 1 cos a - b = cos a cos b + sin a sin b - 2 Subtracting equation 1 from 2, we have cos a - b - cos a + b = cos a cos b + sin a sin b - cos a cos b - sin a sin b ⇒ cos a - b - cos a + b = cos a cos b + sin a sin b - cos a cos b + sin a sin b ⇒ cos a - b - cos a + b = cos a cos b - cos a cos b + sin a sin b + sin a sin b ⇒ cos a - b - cos a + b = sin a sin b + sin a sin b [The term cos a cos b got cancelled because of opposite signs] ⇒ cos a - b - cos a + b = 2 sin a sin b ⇒ sin a sin b = 1/2[cos a - b - cos a + b] Hence the sina sinb formula has been derived. Thus, sina sinb = 1/2[cosa - b - cosa + b] How to Apply Sina Sinb Formula? Next, we will understand the application of sina sinb formula in solving various problems since we have derived the formula. The sin a sin b identity can be used to solve simple trigonometric problems and complex integration problems. Let us go through some examples to understand the concept clearly and follow the steps given below to learn to apply sin a sin b identity Example 1 Express sin x sin 7x as a difference of the cosine function using sina sinb formula. Step 1 We know that sin a sin b = 1/2[cosa - b - cosa + b]. Identify a and b in the given expression. Here a = x, b = 7x. Using the above formula, we will proceed to the second step. Step 2 Substitute the values of a and b in the formula. sin x sin 7x = 1/2[cos x - 7x - cos x + 7x] ⇒ sin x sin 7x = 1/2[cos -6x - cos 8x] ⇒ sin x sin 7x = 1/2 cos 6x - 1/2 cos 8x [Because cos-a = cos a] Hence, sin x sin 7x can be expressed as 1/2 cos 6x - 1/2 cos 8x as a difference of the cosine function. Example 2 Solve the integral ∫ sin 2x sin 5x dx. To solve the integral ∫ sin 2x sin 5x dx, we will use the sin a sin b formula. Step 1 We know that sin a sin b = 1/2[cosa - b - cosa + b] Identify a and b in the given expression. Here a = 2x, b = 5x. Using the above formula, we have Step 2 Substitute the values of a and b in the formula and solve the integral. sin 2x sin 5x = 1/2[cos 2x - 5x - cos 2x + 5x] ⇒ sin 2x sin 5x = 1/2[cos -3x - cos 7x] ⇒ sin 2x sin 5x = 1/2cos 3x - 1/2cos 7x [Because cos-a = cos a] Step 3 Now, substitute sin 2x sin 5x = 1/2cos 3x - 1/2cos 7x into the intergral ∫ sin 2x sin 5x dx. We will use the integral formula of the cosine function ∫ cos x = sin x + C ∫ sin 2x sin 5x dx = ∫ [1/2cos 3x - 1/2cos 7x] dx ⇒ ∫ sin 2x sin 5x dx = 1/2 ∫ cos 3x dx - 1/2 ∫ cos 7x dx ⇒ ∫ sin 2x sin 5x dx = 1/2 [sin 3x]/3 - 1/2 [sin 7x]/7 + C ⇒ ∫ sin 2x sin 5x dx = 1/6 sin 3x - 1/14 sin 7x + C Hence, the integral ∫ sin 2x sin 5x dx = 1/6 sin 3x - 1/14 sin 7x + C using the sin a sin b formula. Important Notes on sina sinb Formula sin a sin b is applied when either the two angles a and b are known or when the sum and difference of angles are known. sin a sin b = 1/2[cosa - b - cosa + b] It can be derived using angle sum and difference identities of the cosine function Topics Related to sina sinb cos a cos b cos 2pi cos a - b FAQs on Sina Sinb What is Sina Sinb Formula in Trigonometry? Sina Sinb is an important formula in trigonometry that is used to simplify various problems in trigonometry. The sin a sin b formula is sin a sin b = 1/2[cosa - b - cosa + b]. What is the Formula of 2 Sina sinb? We know that sina sinb = 1/2[cosa - b - cosa + b] ⇒ 2 sin a sin b = cosa - b - cosa + b. Hence the formula of 2 sin a sin b is cosa - b - cosa + b. How to Prove sina sinb Identity? The trigonometric identities which are used to derive the sina sinb formula are cos a + b = cos a cos b - sin a sin b cos a - b = cos a cos b + sin a sin b Subtract the above two equations and simplify to derive the sin a sin b identity. What is the Expansion of Sina Sinb in Trigonometry? The sina sinb expansion formula in trigonometry for angles a and b is given as, sin a sin b = 1/2[cosa - b - cosa + b]. Here, a and b are angles, and a + b and a - b are their compound angles. How to Apply Sina Sinb Formula? The sina sinb identity can be used to solve simple trigonometric problems and complex integration problems. The formula for sin a sin b can be applied in terms of cos a - b and cos a + b to solve various problems. How to Use sina sinb Identity in Trigonometry? To use sin a sin b formula, compare the given expression with the formula sin a sin b = 1/2[cosa - b - cosa + b] and substitute the corresponding values of angles a and b to solve the problem.