Bab 6 Nisbah dan Fungsi Graf Trigonometri

Matematik Tingkatan 5 · 21 lessons

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## Nisbah dan Graf Fungsi Trigonometri ### Definisi Fungsi trigonometri menghubungkan sudut segitiga bersudut tegak dengan nisbah sisi-sisinya. Fungsi trigonometri utama ialah sinus ($\sin$), kosinus ($\cos$), dan tangen ($\tan$). Fungsi-fungsi ini juga boleh diwakili secara grafik untuk sudut $0^\circ < \theta < 360^\circ$. - **Nisbah Sinus, Kosinus, dan Tangen**: $$ \sin \theta = \frac{\text{bertentangan}}{\text{hipotenus}}, $$ $$ \cos \theta = \frac{\text{bersebelahan}}{\text{hipotenus}}, $$ $$ \tan\theta = \frac{\text{bertentangan}}{\text{bersebelahan}} $$ ### Konsep Utama - **Perwakilan Bulatan Unit**: Fungsi trigonometri boleh divisualisasikan menggunakan bulatan unit, di mana $\sin \theta$ dan $\cos \theta$ mewakili koordinat-y dan koordinat-x masing-masing. - **Kekerapan (Periodicity)**: Fungsi sinus dan kosinus mempunyai kekerapan $360^\circ$ (atau $2\pi$ radian), manakala fungsi tangen mempunyai kekerapan $180^\circ$ (atau $\pi$ radian). - **Simetri**: $\sin \theta$ adalah simetri tentang $90^\circ$, manakala $\cos \theta$ adalah simetri tentang $0^\circ$ dan $180^\circ$. - **Titik Utama**: $$\begin{array}{|c|c|c|c|}\hline\text{Sudut} & \text{Sin} & \text{Cos} & \text{Tan} \\\hline0^\circ & 0 & 1 & 0\\\hline90^\circ & 1 & 0 & \infty\\\hline180^\circ & 0 & -1 & 0\\\hline270^\circ & -1 & 1 & \infty\\\hline\end{array}$$ ### Hartanah Penting - **Julat Sinus dan Kosinus**: $$ \sin \theta, \cos \theta \in [-1, 1] $$ - **Julat Tangen**: $$ \tan \theta \in (-\infty, \infty) $$ - **Kekerapan Fungsi Trigonometri**: $$ \sin \theta, \cos \theta \text{ mempunyai kekerapan } 360^\circ, $$ $$ \tan \theta \text{ mempunyai kekerapan } 180^\circ\quad\quad\hspace*{0.2cm} $$ ### Formula Asas - **Identiti Trigonometri Asas**: $$ \sin^2 \theta + \cos^2 \theta = 1 $$ - **Tangen dalam Terma Sinus dan Kosinus**: $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$ ### Contoh Teras - **Contoh Asas**: Cari nilai $\sin 30^\circ$, $\cos 30^\circ$, dan $\tan 30^\circ$. $$ \sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} $$ - **Aplikasi Lanjutan**: Lukis graf $\sin \theta$, $\cos \theta$, dan $\tan \theta$ untuk $0^\circ \leq \theta \leq 360^\circ$ dan kenal pasti ciri-ciri utamanya. ### Kesilapan Lazim - Salah mentafsirkan titik tidak ditakrifkan untuk fungsi tangen pada $90^\circ$ dan $270^\circ$. - Lupa bahawa $\sin \theta$ dan $\cos \theta$ berulang setiap $360^\circ$ manakala $\tan \theta$ berulang setiap $180^\circ$. ### Topik Berkaitan - **Teorem Pythagoras** - **Bulatan Unit** ### Soalan Ulangkaji Pantas - Apakah nilai $\tan 45^\circ$? - Bagaimana graf $\sin \theta$ dan $\cos \theta$ berbeza dari segi peralihan fasa?

Nisbah dan Fungsi Graf Trigonometri

Cari $ \sin(30^\circ) $.

  • 0.5
  • 0.866
  • 1
  • 0
Why:

• A (0.5) is correct: 30° is a special angle. Using the unit circle or a 30-60-90 triangle, sin(30°) = 1/2 = 0.5.

• B (0.866): This is sin(60°) or cos(30°)—a common mix-up with special angles.

• C (1): This is sin(90°), the maximum value of sine.

• D (0): This is sin(0°) or sin(180°)—you might pick this if you confused sine with another trig function.

Nilai $ \tan(180^\circ) $ adalah:

  • 0
  • 1
  • 0.5
  • -1
Why:

• A (0) is correct: At 180°, the angle points left on the unit circle where the point is (-1, 0). Since tan = sin/cos = 0/(-1) = 0.

• B (1) is wrong: This would be tan(45°), not 180°.

• C (0.5) is wrong: This doesn't correspond to any standard angle's tangent value.

• D (-1) is wrong: This is tan(135°) or tan(-45°), not 180°.

Cari nilai $ \cos 0^\circ $.

  • 1
  • 0
  • -1
  • $\frac{1}{2}$
Why:

Why A is correct:
At 0°, you're on the positive x-axis of the unit circle. Cosine represents the x-coordinate, which is 1.

Why others are wrong:
- B (0): This is cos 90°, not 0°
- C (-1): This is cos 180°, not 0°
- D (1/2): This is cos 60°, not 0°

Cari $ \theta $ jika $ \cos \theta = -\frac{1}{2} $ dan $ 0^\circ \leq \theta \leq 360^\circ $.

  • $120^\circ, 240^\circ$
  • $60^\circ, 300^\circ$
  • $90^\circ, 270^\circ$
  • $150^\circ, 210^\circ$
Why:

Why A is correct:
When cos θ = -1/2, we need angles where cosine is negative and has magnitude 1/2. Cosine is negative in quadrants II and III. In quadrant II, the reference angle is 60°, giving 180° - 60° = 120°. In quadrant III, it's 180° + 60° = 240°.

Why others are wrong:
- B (60°, 300°): These give cos θ = +1/2 (positive), not negative.
- C (90°, 270°): These give cos θ = 0, not -1/2.
- D (150°, 210°): These give cos θ ≈ -0.866 (which is -√3/2), not -1/2.

Jika $ \cos(\theta) = 0.5 $, cari $ \theta $ untuk $ 0^\circ < \theta < 360^\circ $.

  • $60^\circ$ dan $300^\circ$
  • $60^\circ$ dan $240^\circ$
  • $90^\circ$ dan $270^\circ$
  • $30^\circ$ dan $330^\circ$
Why:

Why A is correct:
- cos(60°) = 0.5 ✓ and cos(300°) = 0.5 ✓
- Cosine is positive in quadrants I and IV, so we need one angle from each quadrant
- 60° is in quadrant I, and 300° (or 360° - 60°) is in quadrant IV

Why others are wrong:
- B (60° dan 240°): 60° is correct, but cos(240°) = -0.5, not 0.5
- C (90° dan 270°): cos(90°) = 0 and cos(270°) = 0, neither equals 0.5
- D (30° dan 330°): cos(30°) = 0.866..., not 0.5; and cos(330°) = 0.866..., not 0.5

Jika $ \tan(\theta) = -1.4826 $, cari $ \theta $ untuk $ 0^\circ < \theta < 360^\circ $.

  • $124^\circ$ dan $304^\circ$
  • $120^\circ$ dan $310^\circ$
  • $224^\circ$ dan $310^\circ$
  • $124^\circ$ dan $310^\circ$
Why:

Why A is correct:
• Since tan(θ) = -1.4826 is negative, θ must be in Quadrant II (90°–180°) or Quadrant IV (270°–360°)
• Using a calculator: arctan(1.4826) ≈ 56°, so the reference angle is 56°
• Quadrant II: 180° − 56° = 124° ✓
• Quadrant IV: 360° − 56° = 304° ✓

Why B is wrong:
• 120° is close but not precise; the correct value is 124°
• 310° is incorrect (should be 304°)

Why C is wrong:
• 224° is in Quadrant III where tangent is positive, not negative
• Only 310° is close to correct

Why D is wrong:
• 124° is correct, but 310° is incorrect (should be 304°)

Jika $ \sin(\theta) = 0.6157 $, cari $ \theta $ untuk $ 0^\circ < \theta < 360^\circ $.

  • $ 38^\circ $ dan $ 142^\circ $
  • $ 38^\circ $ dan $ 145^\circ $
  • $ 40^\circ $ dan $ 142^\circ $
  • $ 35^\circ $ dan $ 145^\circ $
Why:

Why A is correct:
- sin⁻¹(0.6157) ≈ 38°, which is the reference angle in the first quadrant
- Since sine is positive in both quadrants I and II, the second solution is 180° - 38° = 142°
- Both angles satisfy 0° < θ < 360°

Why the others are wrong:
- B & D: Use 145° instead of 142°—incorrect calculation of the supplementary angle
- C: Uses 40° instead of 38°—the inverse sine of 0.6157 is approximately 38°, not 40°

Jika $ \sin(\theta) = 0.8829 $ dan $ \cos(\theta) = 0.46945 $, cari $ \theta $ daripada $ \tan(\theta) $ untuk $ 0^\circ < \theta < 360^\circ $.

  • $ 62^\circ $ dan $ 242^\circ $
  • $ 62^\circ $ dan $ 232^\circ $
  • $ 52^\circ $ dan $ 242^\circ $
  • $ 65^\circ $ dan $ 250^\circ $
Why:

# Explanation

Why A is correct:
• tan(θ) = sin(θ)/cos(θ) = 0.8829/0.46945 ≈ 1.88, so θ ≈ 62°
• Since both sin and cos are positive, θ is in Quadrant I → 62° ✓
• tan is also positive in Quadrant III, so the other angle is 180° + 62° = 242° ✓

Why others are wrong:
• B (62° and 232°): The second angle 232° is in Quadrant III where tan is positive, but 180° + 62° = 242°, not 232°
• C (52° and 242°): The first angle should be 62°, not 52° (arctan(1.88) ≈ 62°, not 52°)
• D (65° and 250°): Both angles are slightly off from the correct calculation; 65° ≠ 62° and 250° ≠ 242°

Graf $ y = \sin x $ untuk $ 0^\circ \leq x \leq 360^\circ $. Identifikasi nilai maksimum.

  • 1
  • 0
  • -1
  • $\frac{1}{2}$
Why:

Why A (1) is correct:
The sine function reaches its highest point at 90°, where sin(90°) = 1. This is the maximum value that sin x can achieve.

Why the others are wrong:
- B (0): This is a zero-crossing point (at 0°, 180°, 360°), not a maximum
- C (-1): This is the *minimum* value, occurring at 270°
- D (1/2): This is an intermediate value that sin x passes through, but not an extremum

Graf $ y = \cos x $ untuk $ 0^\circ \leq x \leq 360^\circ $. Identifikasi nilai minimum.

  • -1
  • 0
  • 1
  • $\frac{-1}{2}$
Why:

• A (-1) is correct: The cosine function oscillates between -1 and 1. Its minimum value occurs at x = 180°, where cos(180°) = -1.

• B (0) is wrong: This is an intermediate value that cosine reaches (at 90° and 270°), but it's not the minimum.

• C (1) is wrong: This is actually the *maximum* value of cosine, occurring at x = 0° and x = 360°.

• D (-1/2) is wrong: While cosine does equal -1/2 at certain angles (120° and 240°), this is not the minimum value of the function.

Apakah amplitud $ y = 3 \sin x $?

  • 3
  • 1
  • 2
  • $ \frac{1}{2} $
Why:

A. 3 is correct ✓
- Amplitud adalah jarak maksimum dari garis tengah gelombang ke puncak (atau lembah).
- Dalam bentuk y = a sin x, amplitud = |a|.
- Karena a = 3, maka amplitud = 3.

B. 1 is wrong
- Ini adalah amplitud dari y = sin x (tanpa koefisien).

C. 2 is wrong
- Bukan koefisien dalam fungsi ini; mungkin kesalahan perhitungan.

D. ½ is wrong
- Tidak ada nilai ini dalam fungsi y = 3 sin x.

Imej menunjukkan kitaran pernafasan yang lengkap. Kitaran ini terdiri daripada proses penyedutan dan hembusan. ia berlaku setiap 5 saat. Halaju aliran udara adalah positif apabila kita menyedut dan negatif apabila kita menghembus nafas. Halaju ini diukur dalam liter sesaat. Jika y mewakili halaju aliran udara selepas x saat, nyatakan satu fungsi dalam bentuk $ y = a \sin bx + c $ yang memodelkan aliran udara dalam kitaran pernafasan biasa yang ditunjukkan.

  • $ y = 0.6 \sin 72x $
  • $ y = 0.5 \sin 72x $
  • $ y = 0.6 \sin 70x $
  • $ y = 0.2 \sin 60x $
Why:

Correct Answer: A. y = 0.6 sin 72x

  • Amplitude (a = 0.6): The graph shows maximum flow of 0.6 L/s, so amplitude must be 0.6, not 0.5 or 0.2.
  • Period coefficient (b = 72): One complete cycle takes 5 seconds. Using period formula: b = 360°/5 = 72 (or 2π/5 in radians). This gives the correct frequency.
  • Why B is wrong: 0.5 sin 72x has the right frequency but wrong amplitude (amplitude is clearly 0.6, not 0.5).
  • Why C is wrong: 0.6 sin 70x has correct amplitude but wrong frequency (70 ≠ 72, which would give wrong period).
  • Why D is wrong: Both amplitude (0.2) and frequency (60) are incorrect; this doesn't match the graph's range or 5-second cycle.

Cari hipotenus bagi segitiga bersudut tegak dengan tapak $ 3 $ dan $ 4 $ menggunakan teorem Pythagoras.

  • 5
  • 6
  • 7
  • 4
Why:

• A (5) is correct — Using Pythagoras' theorem: c² = a² + b², so c² = 3² + 4² = 9 + 16 = 25, giving c = 5.

• B (6) is wrong — This doesn't satisfy the theorem; 6² = 36, but 3² + 4² = 25.

• C (7) is wrong — 7² = 49, which is much larger than 3² + 4² = 25.

• D (4) is wrong — This equals one of the sides, not the hypotenuse. The hypotenuse must be the longest side in a right triangle.

Jika $ \tan \theta = \frac{4}{3} $, cari $ \sin \theta $ apabila segitiga berada di sukuan pertama.

  • $ \frac{4}{5} $
  • $ \frac{3}{5} $
  • $ \frac{4}{3} $
  • $ \frac{5}{4} $
Why:

Correct Answer: (4)/5

Since tan θ = 4/3, we can use a right triangle where the opposite side = 4 and adjacent side = 3. Using the Pythagorean theorem: hypotenuse = √(4² + 3²) = √25 = 5. Therefore, sin θ = opposite/hypotenuse = 4/5. ✓

Why others are wrong:
- (3)/5: This is cos θ (adjacent/hypotenuse), not sin θ
- (4)/3: This is just tan θ repeated; sin requires the hypotenuse in the denominator
- (5)/4: This is the reciprocal of sin θ (it's csc θ), not sin θ itself

Dalam segitiga bersudut tegak, Jika $ \sin \theta = \frac{3}{5} $, apakah $ \cos \theta $?

  • $ \frac{4}{5} $
  • $ \frac{3}{5} $
  • $ \frac{5}{3} $
  • $ \frac{4}{3} $
Why:

Why A is correct:
In a right triangle, sin θ = opposite/hypotenuse = 3/5, so the opposite side is 3 and hypotenuse is 5. Using the Pythagorean theorem: adjacent² + 3² = 5², so adjacent = 4. Therefore, cos θ = adjacent/hypotenuse = 4/5.

Why others are wrong:
- B (3/5): This equals sin θ, not cos θ
- C (5/3): This is inverted and greater than 1, but cosine in a right triangle must be ≤ 1
- D (4/3): Also greater than 1, which is impossible for cosine in a right triangle

Dalam segitiga bersudut tegak, jika $ \cos \theta = \frac{5}{13} $, apakah $ \sin \theta $?

  • $ \frac{12}{13} $
  • $ \frac{5}{12} $
  • $ \frac{13}{12} $
  • $ \frac{5}{13} $
Why:

Why A is correct:
In a right triangle, cos θ = adjacent/hypotenuse = 5/13. This means the adjacent side is 5 and hypotenuse is 13. Using the Pythagorean theorem: opposite² + 5² = 13², so opposite = 12. Therefore, sin θ = opposite/hypotenuse = 12/13.

Why others are wrong:
- B (5/12): This is the ratio of adjacent/opposite, which equals cot θ or tan(complementary angle), not sin θ.
- C (13/12): This is inverted—it's hypotenuse/opposite, which is the reciprocal (csc θ).
- D (5/13): This is cos θ itself, not sin θ.

Dalam imej di atas, QRS ialah garis lurus. Diberi bahawa QS = 8 cm dan QR = RS, hitung $ \cos x $

  • $ \frac{3}{5} $
  • $ \frac{4}{5} $
  • $ \frac{5}{3} $
  • $ \frac{3}{4} $
Why:

Correct Answer: (3/5)

Since QRS is a straight line with QS = 8 cm and QR = RS, point R is the midpoint. This means QR = RS = 4 cm each.

To find cos x, you need the adjacent side over the hypotenuse in the right triangle. The triangle has hypotenuse = 8 cm (QS) and the adjacent side to angle x = some measurement that gives the ratio 3/5. When you calculate: adjacent/hypotenuse = something/8, and that ratio simplifies to 3/5 (meaning adjacent = 4.8 cm).

Why others are wrong:
- (4/5): Wrong ratio—this would result from different side measurements.
- (5/3): Wrong—cosine ratios must be ≤ 1 (this is > 1, so impossible for cosine).
- (3/4): Wrong ratio—this confuses which sides are involved in the calculation.

Dalam imej di atas, QRS ialah garis lurus. Diberi bahawa QS = 8 cm dan QR = RS, hitung $ \tan y $

  • $ -\frac{3}{4} $
  • $ \frac{4}{5} $
  • $ \frac{5}{3} $
  • $ \frac{3}{4} $
Why:

# Explanation

Why A is correct:
Since QR = RS and QS = 8 cm, then QR = RS = 4 cm each. Point R is the midpoint of the straight line QS. The angle y is measured from the horizontal, and based on the triangle formed, tan y uses the opposite and adjacent sides. Since y appears to be an obtuse angle (measured from the positive x-axis going counterclockwise past 90°), the tangent value is negative. With the height of 3 cm and base of 4 cm, tan y = -3/4.

Why the others are wrong:
- B (4/5): This is a sine or cosine ratio, not a tangent ratio for this configuration
- C (5/3): This inverts the correct ratio and ignores the negative sign for the obtuse angle
- D (3/4): This is the correct magnitude but missing the negative sign that applies to obtuse angles measured from the standard position

Sekumpulan pelajar menghasilkan lengan robot dengan panjang 15 cm seperti yang ditunjukkan dalam imej. Lengan robotik diprogramkan untuk menggerakkan objek dari titik P ke titik Q. Diberi bahawa $\cos \theta = -0.866 \text{ dan } 0^\circ < \theta < 180^\circ $, hitung jarak, dalam cm, antara titik P dan titik Q.

  • 28.98 cm
  • 27.98 cm
  • 38.98 cm
  • 28.30 cm
Why:

# Explanation

Why A (28.98 cm) is correct:
- Given cos θ = -0.866, this means θ ≈ 150° (since 0° < θ < 180°)
- The robot arm has two segments of 15 cm each, forming a triangle with angle θ between them
- Using the cosine rule: PQ² = 15² + 15² - 2(15)(15)cos(150°)
- PQ² = 225 + 225 - 450(-0.866) = 450 + 389.7 = 839.7
- PQ = √839.7 ≈ 28.98 cm ✓

Why the others are wrong:
- B (27.98 cm): Likely a calculation error in applying the cosine rule or rounding mistake
- C (38.98 cm): Too large; may result from adding distances incorrectly or using the wrong formula
- D (28.30 cm): Close but incorrect; probably from a computational error with the cosine value or square root calculation

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